The defending champion quickly took the first set and was two breaks up in the second when the match had to be halted for the roof to be closed.
The South African fought hard in the third to take the set to a tiebreak, but was unable to avoid the defeat.
Read how all the action unfolded below:
3.40pmGood afternoon! The defending champion Andy Murray will be looking to secure a place in the quarter-finals at Wimbledon when he makes his way at to Centre Court shortly to face 20th seed Kevin Anderson. The start of the match may be pushed back by a few minutes, as the roof, which has been closed following a shower earlier, could be opened before the tie gets underway.
3.43pmWhile Murray has progressed with relative ease so far, Anderson has had a bit of a battle to reach to last 16. He knocked out Italy's Fabio Fognini in straight sets in the third round, following a four-set encounter in the second round.
3.45pmThe roof is now being opened, so it will probably be about 5-10 minutes before the players make their way out. Hopefully this means the weather will be good for the rest of the afternoon.
3.47pmThis is just the third time that these two have met on the ATP tour, and their first match since 2011. In their last meeting, it was the South African who took the victory, beating Murray in straight sets in Montreal. The Scot won their only other match, which was at the 2010 Australian Open, but they've never played on grass before.
3.48pmThe roof is still in the process of opening, but the players have just come onto the court ready for their warm-up. As ever, the crowd welcome the Brit with a huge cheer. It looks like there are plenty of fans in for the defending champion, with British and Scottish flags around the court.
3.51pmThis is the first time that Anderson has ever played on Centre Court. He admitted that he's been practising the walk out to the court and taking pictures of the corridor from the changing rooms to the showcourt so that he knew which way he would be going.
3.53pmOn paper, this should be a much sterner test than Murray has faced in the previous rounds. The South African is known for his big serve, but has never reached the quarter-final at a Grand Slam.
3.55pmSince he missed the tournament with injury in 2007, Murray has reached at least the quarter-finals every year, and has been in the final for the last two.
3.56pmRight then, the warm-ups are over and the players are ready to go, with Anderson to serve first.
Murray *0-1 AndersonAnderson's big serve catches Murray out straight away as the 20th seed claims the opening point of the match. When Murray does manage to get his return in, he gets the better of his opponent with some smart play from the baseline, but Anderson produces another couple of powerful serves out wide to see out the game.
Murray 1-1* AndersonAnderson can't get his return in on the opening two points of the game before Murray fires an ace down the centre of the court. The South African gets his racquet to the ball at 40-0 down, but his return is wide.
Murray *2-1 AndersonBREAK! Murray takes a 30-0 lead after Anderson slices his forehand into the net. The South African tries to get back into the game by coming into the net, but his volley hits the top of the net chord and goes wide to give Murray three break points. Anderson manages to save the first when he forces a long return from Murray, before the second is hit into the net, but the 20th seed smashes his forehand long while trying to take the game to deuce to hand Murray the break.
Murray 3-1* AndersonMurray consolidates his break despite a tight service game. He misses a couple of his first serves, but his second is enough to challenge Anderson. At 40-15, a slice into the net from the defending champion before Anderson pulls it back to deuce, but the Brit ups the power on his serve to see out the game.
Murray *3-2 AndersonAnderson's powerful serve quickly helps him to a 40-15 lead, but Murray keep himself in the game with a forehand winner. However, he can't pick up the next point, slicing his backhand into the net. Meanwhile, there's a light shower starting to come down again, with the groundstaff starting to ready themselves in case the court has to be covered.
Murray 4-2* AndersonThis time it's a relatively straight-forward hold for Murray as he gets the better of Anderson in a rally, coming inside the baseline to volley past his opponent, before wrapping the game up with an ace.
Murray *4-3 AndersonAnderson's first ace of the match helps him to a 30-0 lead, but Murray continues to get the better of him at the net as he levels the scores before pulling off a superb cross-court forehand return to give himself a chance for another break. Anderson slips while stretching for an overhead lob, but Murray's reply heads into the net to take the game to deuce. The defending champions makes his second unforced error of the set to give Anderson advantage, before a powerful serve from the 20th seed sealed the game. Murray remains a break up though.
Murray 5-3* AndersonMurray finds himself behind when Anderson catching his out with a deep return from a drop shot, but he levels with his fourth ace of the match. At deuce, another ace gives the third seed advantage before Anderson misses with a forehand. The South Africa battled hard to try and get the set back on serve, but every time it looked like he could edge ahead, Murray responded with a powerful serve. It looks like the Brit has plenty in the tank should Anderson up his game.
Murray *5-4 AndersonA 128mph ace from Anderson takes him to a 40-15 lead, but a long forehand from the 20th seed before a cross-court forehand into the sideline from Murray takes the game to deuce, but Anderson did enough to hold serve, leaving Murray to try and serve the set out next.
4.42pmAfter 43 minutes, Murray has clinched the first set. The Scot serves superbly to take a 40-15 lead, but can't wrap it up on the first set point as he forces his forehand long, but the South African misses his return on the next point, giving Murray the set.
Murray *1-0 AndersonBREAK! Murray backs a bright start to the set as he returns well to clinch the first two points before Anderson completely mis-hits a smash, firing it into the net from just a couple of metres away to give Murray three break points. He saves the first when Murray fires long, before picking out the baseline with a superb forehand, repeating the shot on the next point to bring up deuce. The Scot gets another chance when the 20th seed's forehand drifts wide, but the game heads back to deuce when the third seed mis-hits a slice. With his seventh break point of the game, Murray finally seals the game when Anderson comes in for a volley, but he misses the sideline.
Murray 2-0* AndersonAnderson battles hard to put Murray under pressure as he tries to get back into this match, but Murray moves well behind the baseline before sealing the game with an ace. Once again, there are a couple of raindrops falling into the court, but not enough to stop play for the time being.
Murray *3-0 AndersonBREAK! Murray accidentally slices a volley into his opponent's chest, but it doesn't seem to faze Anderson as he fires a couple of big serves past his opponent. Murray fights back with a string of shots from behind the baseline to take the game to deuce, despite questioning the umpire over the rain. The defending champion gets the advantage when Anderson fires long before the South African slices into the net on the next point.
5.03pmPLAY SUSPENDED! After checking the court conditions at the change of ends, the umpire has decided the suspend play because of the continuing rain falling onto the court.
5.05pmThe players have left the court and the majority of the spectators are reaching for their umbrellas and rain jackets. The Wimbledon Twitter feed has confirmed that the roof will be closed, so hopefully they'll be able to return as soon as possible.
5.08pmWhile we wait for the action to get back underway, let's look at some of the stats from the match so far. Murray will obviously be the happier of the two as he leads by a set and a double break. The Brit has six aces to Anderson's two. The third seed has won 81% of the point off his first serve, while Anderson has claimed just half.
5.10pmThe roof is nearly closed now so they'll be a bit longer to wait while the conditions are adjusted for indoor play, and then the players should be back out.
5.12pmLooking back at the stats so far, Murray has hit 23 winners and made just three unforced errors, while Anderson has hit 14 winners and 10 unforced errors. Anderson really hasn't done a lot wrong, but Murray has just been a different level so far.
5.15pmPlay has also been suspended over on court one, where Grigor Dimitrov is taking on Leonardo Mayer. The winner of that match faces the winner of this clash, and at the moment it looks like it will be Dimitrov, as he's a set up.
5.19pmIf you've missed any of the results from earlier today, you can catch up with all of the completed ties so far on our Wimbledon results page.
5.21pmGood news, the players are now back on court. As they've been off for a few minutes they'll probably be given a couple of minutes to warm back up, and then the action will be back underway.
5.26pmOne the action does resume, it will be Murray serving first under the roof, having already broken Anderson twice in the second set.
5.28pmRight then, the players have had a warm-up and are ready to continue this fourth-round encounter on Centre Court. It's been 26 minutes since a point was last played.
Murray 6-4 3-1* AndersonBREAK! Murray gets off to a good start as he picks up the first point from a lengthy rally, but hits his forehand into the net off the next point. A couple of mistakes from Murray give the South African his first two break points of the match, and he saves the first with a huge serve, but Anderson gets a lucky hit off the net chord to take the game. The defending champion is still a break up though.
Murray 6-4 *3-2 AndersonMurray immediately fights to regain his two-break cushion, making some solid returns to win the opening couple of points, but Anderson serves well to level. The 20th seed tries to volley, but the Scot sprints quickly to the net to get it back over and give himself another break point. However, he mis-hits his forehand to brings up deuce. After saving one Anderson advantage, Murray hits back-to-back long forehands to allow his opponent to hold serve for the first time in this set.
Murray 6-4 4-2* AndersonAt 30-30, a forehand from Murray finds the net, giving Anderson a chance to break back and level the set, but the South African makes the same mistake himself on the next point to take the game to deuce. The defending champion takes advatange with an ace before sealing the game with a cross-court forehand winner, and looks extremely relieved to have avoided the break.
Murray 6-4 *4-3 AndersonAnderson has easily his best service game of the second set, with Murray failing to return two of his serves, while the South African wraps up the other two points with some clever shots at the net. The 20th seed has certainly looked a lot sharper since the rain delay.
Murray 6-4 5-3* AndersonFor the first time since the match resumed with the roof shut over Centre Court, Murray manages to comfortably hold serve as Anderson fires a couple of long forehands past the Brit before slicing his backhand wide. A long forehand from Murray briefly delays the inevitable, as he wraps the game up with a solid forehand winner on the next point to leave himself just one game away form a two-set lead.
5.55pmAnderson initially seems to be in control as he takes a 30-0 lead, but the errors creep back in to cost the South African, and he gifts Murray the set attempting what should have been a simple volley, but his efforts fails to clear the net. Murray certainly hasn't been able to sweep Anderson aside as he did with his previous three opponents at this year's tournament, but it's been relatively straight-forward so far.
Murray 6-4 6-3 1-0* AndersonMurray picks out the sideline with a superb backhand before firing another ace past his opponent to quickly take a 40-0 lead. Anderson comes inside the baseline to attack his opponent's serve, but Murray has too much power and his return goes well long.
Murray 6-4 6-3 *1-1 AndersonWell it's a comfortable hold for Anderson in his first service game of the third set. He powers an ace past his opponent before making a stunning forehand winner on his way to a hold to love.
Murray 6-4 6-3 2-1* AndersonA forehand mistake from Murray allows Anderson to edge ahead in the game, but he quickly levels with an ace before the 20th seed mis-hits his return, allowing the defending champion to wrap up the game with an overhead smash at the net on the next point.
Murray 6-4 6-3 *2-2 AndersonAfter a couple of mistakes from Murray, an ace helps Anderson to a 40-0 lead. The third seed tries to get back into it with a backhand smash, but hits the net on the next point to allow Anderson to hold serve.
Murray 6-4 6-3 3-2* AndersonMurray quickly notches up another comfortable hold, making one of the shots of the match on his way as he stretches out wide to return an Anderson forehand cross-court into the back corner.
Murray 6-4 6-3 *3-3 AndersonMurray produces a fantastically placed drop shot right on the sideline as he tries to get a break in this third set, but Anderson manager to edge ahead before the Brit hits a forehand just long to give him the game. Incidentally, Murray challenged the call on the long forehand, which was the first time that Hawkeye had been called upon all match.
Muuray 6-4 6-3 4-3* AndersonMurray will be a bit relieved to have held that game, as Anderson did take the lead when the third seed served a double fault, but he responded well to see out the game with an ace.
Murray 6-4 6-3 *4-4 AndersonMurray has five break points in a lengthy battle with the South African, but each time Anderson relies on his serve to save it before eventually forcing advantage and taking the game when Murray fires his return long.
Murray 6-4 6-3 5-4* AndersonMurray takes a 40-15 lead as he tries to race through his own service game, but he makes his second double fault of the match to give Anderson a chance of coming back. However, the 20th seed goes too big on his return looking for the next point to let the third seed hold. He needs just one more game to reach the quarter-finals, but is yet to break Anderson in this set.
Murray 6-4 6-3 *5-5 AndersonMurray takes a 30-15 lead and tries to bring up match point with an ambitious backhand slice, but it drifts just wide, before Anderson takes the lead with a volley and sees the game out with a big serve to keep the set level.
Murray 6-4 6-3 6-5* AndersonMurray starts the game with an ace before rifling a couple of forehand winners past his opponent. Anderson gets a point back with a solid backhand, but can't repeat the move as he slices his shot into the net on the next point. Murray will have at least a tiebreak from this set.
Murray 6-4 6-3 6-6 AndersonAnderson blasts the ball well long in the opening point of the game, but hits a forehand and volley winner to get himself back in the game before Murray hits long to take the set to a tiebreak.
TIEBREAK!Murray *5-6 Anderson. Anderson has a chance to force a fourth set!
TIEBREAK!Murray *6-6 Anderson. However, his return goes well long
TIEBREAK!Murray 7-6* Anderson. Anderson fires long again to give Murray his first match point.
6.56pmGAME, SET & MATCH! Andy Murray 6-4 6-3 7-6 Kevin Anderson
6.57pmAnderson battled bravely in the tiebreak to try and fourth the match to a fourth set, but Murray was just too strong as he secures his place in the quarter-finals for a seventh year in a row. It's certainly the toughest match Murray has faced at Wimbledon this year, but he's safely through.
6.59pmWell that's all from Centre Court for now, but stick around as we'll have live coverage of Novak Djokovic's match against Jo-Wilfried Tsonga shortly, which should be a highly entertaining encounter. Thanks for joining me this evening, goodbye.