The British teenager started brightly and served for the first set at 5-2 ahead, but eventually lost out to the Estonian in a tense tie-breaker.
One break of serve proved decisive in the second set and it went Kanepi's way, ending Robson's tournament in the process in a 6-7(6) 5-7 defeat.
Look back at a gutsy effort from the 19-year-old with Sports Mole's coverage below.
12.57pmAfternoon tennis fans. The second week of Wimbledon marks the start of the serious stuff at SW19, and for the first time since 1998 there British interest in both the men's and women's singles draw. Laura Robson is in action first this afternoon, against dangerous Estonian Kaia Kanepi.
12.59pmRobson's run to the fourth round has really captured the imagination of the British public, with the 19-year-old building momentum and confidence after the early scalp of Maria Kirilenko. Another stern test awaits today though on Court One against Kanepi, who has reached the quarter-finals at SW19 once before.
1.02pmBoth players are due on Court One in just a few moments, in what on paper looks to be a fairly even encounter. There are just eight places between the two girls in the world rankings, with Kanepi perhaps having a slight edge in terms of experience. Robson has been fearless in her approach so far though, and a similar gameplan should suit her well today.
1.05pmA huge cheers greets Robson as both players arrive on Court One, which looks to be packed for this opening match of week two at SW19. At her home Grand Slam the 19-year-old has definitely fed off the energy of the crowd, attacking her shots early under pressure.
1.08pmAs both women warm up there is just time to give you a closer look at Kanepi. The Estonian has reached 15 in the world, using a powerful serve to dominate lesser opponents. Injury has blighter her progress this season, but Kanepi has looked back to something approaching her best in the past seven days.
1.10pmThe bookies make Kanepi a slight favourite to progress this afternoon, with both players fighting for what looks likely to be a meeting with Serena Williams in the last eight. Robson has upset the odds more than once in her run to the fourth round though and needs to start quickly, using that left-handed serve to cause her opponent problems at the baseline. This has all the makings of a very competitive, entertaining clash.
1.12pmHere we go then on Court One, with Kanepi to serve first. Important for the teenager Robson to start fast in this match, with Kanepi definitely up there with the hardest hitters on tour.
1.15pmRobson finds her range with a superb forehand winner, levelling the opening game at 30-30 after punishing a sloppy second serve. The Estonian recovers but quickly shows nerves with her first double fault of the match, sending us to deuce.
Robson *0-1 KanepiNice early pressure and aggression from Robson, but an ace followed by a strong forehand down the line just about prove enough to earn Kanepi the early hold.
Robson 1-1* KanepiA promising start on serve for Robson, putting a kick behind her first serve to quickly rattle off her opening service game to love. The Court One crowd thoroughly enjoyed that game from the teenager.
1.19pmBREAK POINT! The Estonian smashes a second ace down the line, with another sign of her power at the baseline following with a winner in the next point. Robson fights right back with an aggrssive play, attacking the net before launching a backhand winner across court for break point.
Robson *1-2 KanepiRobson just spurns a chance to take charge of the first set, with Kanepi going back to basics behind a first serve to get the hard-fought hold. The Brit is clearly happy to exchange with her powerful opponent at the baseline though, which does well for a match full of attacking intent.
Robson 2-2* KanepiA first ace from Robson up the middle, who has definitely worked hard on that service action in the last few months. Kanepi then blasts a pair of forehands well past the baseline, allowing the Brit to claim another comfortable hold.
Robson *2-3 KanepiKanepi rushes a pair of serves and looks to be a struggling a touch with her ball toss, giving Robson an opening at 30-30. The Brit fails to capitalise though, showing nerves of her own with a wayward unforced error at the baseline. A challenge from Robson then proves fruitless, giving Kanepi a hold she desperately needed for confidence.
1.30pmBoth players looked a little edgy in that last game, perhaps a sign that the Court One crowd are playing a part. Robson is back on serve in just a moment, looking to keep the pressure on after a confident start to the first set.
Robson 3-3* KanepiMake that a second hold to love for Robson, who is definitely finding her groove with a pair of awesome forehand winners into the corner. Kanepi has failed to put any pressure on the Brit's serve so far, preferring to stay deep at the baseline and wait for errors.
Robson *3-4 KanepiThe first love service game from Kanepi, who starts to get that powerful first serve working with variety and width. Robson went for an ambitious backhand winner down the line in a gusty move, but saw her effort at 30-0 drift just beyond the sideline.
Robson 4-4* KanepiAnother confident display from Robson, who stays calm at 40-30 on second serve to grind out a crucial hold. Kanepi still looks a tad tentative and stubbornly refuses to leave the baseline, with the first set still very much on serve.
1.38pmBREAK POINTS! A huge chance for Robson as she rattles off successive points on Kanepi's serve, earning warm applause from the crowd and a 0-30 lead. Kanepi then hammers a forehand well wide to give the Brit three break points.
Robson *5-4 KanepiBREAK! An awesome forehand from Robson across court earns a vital break at the second time of asking. That was a fearless effort under pressure, with Kanepi unable to compete due to sloppy footwork at the baseline. The 19-year-old will now serve for the first set!
1.41pmRobson serves her first double fault but responds with a well-placed second serve in the next point, levelling at 15-15.
1.44pmBREAK POINT! Kanepi shows her raw power for the first time in the match, earning a 15-40 advantage after clinical play caught Robson out of position at the baseline.
Robson 5-5* KanepiBREAK! And just like that Kanepi has a break back, with Robson's early joy on serve quickly forgotten after a lapse of concentration. This looks a very even encounter now.
Robson *5-6 KanepiA vast improvement from Kanepi on serve has Robson cursing a series of poor returns on second serve. The Brit must regroup quickly, with a hold essential in the next game as she serves to stay in the first set.
1.50pmMuch more like it from Robson at 0-15 down, moving forward at the net to smash a controlled overhead up the line. Kanepi then blasts a return past the line on second serve, giving her opponent a swift 40-15 advantage.
Robson 6-6* KanepiTIE BREAK! A hold from Robson sends this tense first-set to a tie breaker, which starts with a confident move from the Brit at the net for an early 1-0 lead.
1.53pmA thunderous ace out wide helps Robson build a quick 3-0 lead in the breaker, but Kanepi once again responds strongly with an ace of her own to make score 3-2.
1.55pmRobson leads the tie breaker 4-2 at the change of ends, after getting enough depth behind a forehand across court to force a mistake from Kanepi. Can the teenager stay strong on serve to close out this first set?
1.57pmA touch too much on a backhand return from Robson cuts her advantage to 5-3, with a superb first serve from Kanepi under pressure reducing the deficit further at 5-4. Huge pressure now on the next two Robson serves as the breaker approaches crunch time.
1.58pmSET POINT! Groans from the Court One crowd as Robson hands her opponent a set point, following an untimely double fault with a sloppy forehand down the line. A huge forehand return in the next point keeps Robson alive though, once again levelling the breaker at 6-6 to stop the rot. An absorbing first set continues!
Robson *6-7(6) KanepiSET! Nerves finally get the better of Robson as she tries to save a second set point, spraying a wayward forehand wide of the line. Kanepi showed a touch more composure there as both players tightened up, using a fierce first serve to win the crucial points when it mattered. The 19-year-old has it all to do now, but has produced a spirited fightback already in her run to the fourth round.
2.02pmBREAK POINT! Robson needs to recover quickly and build her confidence in the second set and starts well, serving her third ace up the middle. The Brit then gets some welcome help from Kanepi at 15-30 behind, with the Estonian smashing a sloppy forehand into the net. A double fault soon follows though, handing Kanepi break point.
Robson 6-7(6) 1-0* KanepiMuch more like it under pressure from the Brit, serving a superb ace at break-point down. A measured forehand down the line is followed by some stern baseline defence, securing a hold the 19-year-old desperately needed.
2.08pmRobson wastes a challenge after wrongly questioning a baseline call from the line judge, handing Kanepi a 30-0 lead on serve. She quickly attacks again though, sending a fierce forehand return up the line to prevent the Estonian holding to love.
Robson 6-7(6) *1-1 KanepiUnder pressure Kanepi just gets enough venom behind her first serve to earn the hold, despite some welcome signs of a response at the baseline from Robson to win successive points.
Robson 6-7(6) 2-1* KanepiThe Court One crowd welcomes a confident hold from Robson as she shows aggression to step inside and smash a winner past Kanepi. The first serve looked to have recovered in that game, with the Estonian unable to chase down a wide delivery on two occasions.
2.15pmKanepi blasts a forehand long to give Robson a small opening at 30-30. In the next point the Brit's baseline effort is ruled long, with a challenge reversing the decision and forcing the point to be replayed. The tension on Court One is palpable here early in the second set.
Robson 6-7(6) *2-2 KanepiAfter working so hard to force deuce Robson is unable to take advantage, sending successive forehands well wide at the baseline. Kanepi is playing effective, defensive tennis and just forcing enough mistakes from her opponent to stay in charge of the match.
Robson 6-7(6) 3-2* KanepiThe Brit edged ahead once more in the second set, keeping the ball in play consistently to hold serve. Kanepi remains unwilling to leave the baseline, putting the pressure on her opponent to attack.
Robson 6-7(6) *3-3 KanepiSigns of confidence from the Estonian now as she holds to love with minimum fuss. Kanepi looked very convincing behind a slice serve in that game, putting Robson back on serve in just over a minute.
2.24pmBREAK POINT! Robson needs to dig deep here at 30-30, after being caught on the wrong side of the court by a superb forehand return from Kanepi. A sloppy forehand at the net then drifts wide from Robson, giving her opponent a chance at 30-40.
Robson 6-7(6) 4-3* KanepiA gutsy effort from Robson sees the Brit save successive break-point chances with a pair of aces. She then chases down a short shot at the net, showing neat footwork to swot a forehand across court. Kanepi is then unable to keep a forehand return in play, completing a hold for Robson that was imperative to her chances of producing a comeback.
Robson 6-7(6) *4-4 KanepiThis time its Kanepi's turn to respond under pressure, showing resilience at the baseline to turn a tricky 30-30 game her way. Robson is really struggling with her forehand pass at mid-court range, with the net proving a problem as she moved inside.
Robson 6-7(6) 5-4* KanepiConfident stuff on serve from Robson has the Court One crowd roaring in approval. Kanepi clearly moved inside to put pressure on her opponent's serve in that game but to no avail, with the Brit responding strongly to claim a comfortable hold. Kanepi must now serve to keep the second set alive.
Robson 6-7(6) *5-5 KanepiSuperb anticipation at the net from Kanepi gets her service game off to a strong start, with a fierce first serve quickly building a 40-0 lead. A strong backhand return from Robson reduces the deficit, but Kanepi takes advantage of a slip from the Brit to hold serve.
4.38pmBREAK POINTS! Kanepi finds her range with a superb backhand across court, rattling off successive points on Robson's serve. The Brit's touch then deserts her down the line, handing her opponent three break points.
Robson 6-7(6) 5-6* KanepiAnd just when it matters Robson is unable to match her opponent's power. Kanepi's ferocious return sees her break to love and the Estonian will now serve for the match. Robson needs a heroic effort now to keep her tournament alive.
2.42pmNot a good start from Robson in a game she has to win, slicing an unforced error short of the net. The Brit mixes things up in the next point though, forcing Kanepi to skew a baseline effort well wide to level at 15-15.
2.43pmMATCH POINT! Ace number eight earns Kanepi two match points, but Robson saves the first in a gruelling baseline exchange. The 19-year-old is hanging on by a thread on Court One!
2.44pmMatch point number three goes begging for Kanepi, who sends a shocking double fault at least two feet past the line.
2.45pmMake that four match points saved from Robson as her bold attack is rewarded, with a baseline drive just floating inside the line.
Robson 6-7(6) 5-7 KanepiGAME SET AND MATCH! At the fifth time of asking Kanepi's power tells, sending a measured forehand across court to take a straight-sets victory.
2.47pmHuge disappointment for Robson as she leaves Court One close to tears. The 19-year-old can be immensely proud of her efforts at SW19 this year despite today's exit, producing some memorable victories against the odds.
2.50pmKanepi definitely deserved her victory today, staying remarkably calm to equal her best-ever showing at Wimbledon. Serena Williams or Sabine Lisicki will now await the Estonian in the last eight, and she should not be counted out due to some serious power at the baseline.
2.54pmWell that just about does it for British interest in the women's draw, but it was fun while it lasted. Definitely more to come from Laura Robson at SW19 in the very near future you have to believe. Good afternoon.