Chelsea's clinical finishing proved to be the difference on the day, though, and two more from Morata in the second half saw him complete his hat-trick to fire the champions to a convincing victory.
Find out how all of the action unfolded courtesy of Sports Mole's minute-by-minute updates below.
Ten places and five points separate these two sides in the table after five games of the season, with Stoke entering this match in poor form while Chelsea have overcome something of a false start to put themselves amongst the early front-runners - only the two Manchester clubs are above them in the table at the start of the weekend.
Before we take a closer look at all that, though, let's first check out the team news!
You can be sure that Stoke won't be daunted by the prospect of the champions visiting the bet365 Stadium, but with injury problems mixed in with their recent run of results - which has seen them go four games without winning across all competitions - they head into this match as underdogs.
Not many will be expecting the Potters to be embroiled in a relegation battle this season, nor are they likely to challenge for a European place, so a good cup run could have been valuable for the club this season. They will now be forced to wait for the FA Cup, though, having seen their EFL Cup campaign ended in just the third round.
Mark Hughes's side have won just three of their last 16 outings in the league, losing half of those games and drawing five in a run which stretches back to March. Again, they do enough to keep themselves at arm's length of trouble, but last season was a step in the wrong direction for them and Hughes will be desperate to arrest that slide sooner rather than later.
The Potters have taken just two points from the 15 on offer in the month of September, drawing two games and losing three in that time. Looking even further back they have won just one of their last eight and two of their last 14 Premier League fixtures in September, so victory over the champions would be a significant shock today.
Mark Hughes has worked to evolve Stoke's reputation during his time in charge, but this still remains a difficult place to come for the big teams especially.
That victory over Arsenal is Stoke's only win in their last 10 home Premier League games against one of last season's top six, but they rarely make it easy for the visitors and so far this season are yet to taste defeat in front of their own fans. The four points against Arsenal and Manchester United will have delighted Hughes, and anything on top of that today will be another bonus.
The Potters' rather slow start to the season is somewhat understandable, then, but another defeat here could see Stoke end the day in the relegation zone, and Hughes does not have a particularly encouraging record against Chelsea. This will be his 24th meeting with his former club as a Premier League manager, during which time he has lost 15 times and conceded 43 goals - both higher tallies than against any other club.
It is, of course, still too early in the season to draw too many conclusions from the table, but the Manchester clubs have started so strongly that none of the chasing pack will want to fall too far behind in the opening months. A six-point gap by September would be quite a significant one for Chelsea to make up, although they would have time.
Chelsea come into today's match having won five and drawn one of their six games in all competitions since the defeat to Burnley, a run of form which has seen them climb up to third in the Premier League table.
Speaking of Europe, this is the beginning of a huge week for Conte's side, who take on Atletico Madrid in the Champions League on Wednesday - having just agreed a deal to send Diego Costa back there, of course - before facing Manchester City next weekend.
One more point this afternoon would see them breach the 1,800 point mark in the Premier League, becoming only the third team to do that after Arsenal (1,829) and Manchester United (2,034). More importantly, though, it could see them move level on points with the Manchester clubs should City and United slip up in their 3pm matches against Crystal Palace and Southampton respectively.
Another victory today would see them win their opening three Premier League away games in a season for the first time since 2009, when the third triumph in that run was a 2-1 victory at Stoke.
The biggest issue for Conte right now is the discipline of his side, though. David Luiz was sent off during last weekend's stalemate with Arsenal, leaving Chelsea finishing a game against the Gunners with 10 men for the third time in a row as they failed to score at home under the Italian for the first time.
Chelsea have also committed a league-high 68 fouls in the Premier League so far this season, so Conte is clearly looking for his side to make their marks on games and opponents. Maybe Messrs Terry and Costa were the nice guys after all...
The main issue for Stoke heading into today's game was a crisis at centre-back, with Zouma ineligible to play against his parent club and Cameron, Shawcross and Wimmer all sidelined through injury. Hughes's answer to the problem is to switch to three at the back, with Johnson and Pieters filling in alongside Martins Indi.
There are no other changes for the Potters, though, with Johnson and Sobhi the only new faces.
Shaqiri's last six Premier League goals have all come from outside the area, so he will be a threat from range, while Choupo-Moting's brace against Manchester United proved that he also has an eye for goal. Together they have created nine goalscoring opportunities for each other this season - second to only the Eriksen and Kane partnership in the Premier League.
The Belgian was expected to be named in the starting XI this afternoon having started the EFL Cup win over Nottingham Forest in midweek, but despite having three sub appearances in addition to that, Conte does not believe that he is fit enough to start just yet.
The Spaniard has scored three Premier League goals and registered two assists since joining Chelsea, and curiously all of those have come with his head. Stoke are, of course, a physical team, but with a makeshift centre-back unit today Morata may fancy his chances of getting some joy in the air again today.
Luiz is, of course, suspended following his red card against Arsenal last weekend, but Azpilicueta does come back into the team and will wear the armband in the absence of Cahill.
Elsewhere, Pedro has overcome a leg injury to start alongside Willian and Morata in that front three.
Stoke have had some good results against the big teams to have visited here so far this season, and they will be hopeful of coming away from this one with at least another point too. Chelsea still go into the game as favourites, though, and considering Stoke's injury problems I am backing an away win today.
Sports Mole says: Stoke 0-2 Chelsea
Jonathan Walters had earlier cancelled out Willian's opener from the spot, but Cahill ensured that Chelsea left with maximum points on their way to the Premier League title.
Thirteen of their 18 meetings have ended in defeat, though, which is their worst record against any team in the competition.
Cahill and Willian (2) were both on the scoresheet once again on that occasion, with Costa also helping himself to a goal as Chelsea ran out 4-2 winners. Martins Indi and Crouch got the goals for Chelsea.
STOKE STARTING XI: Butland; Johnson, Martins Indi, Pieters; Diouf, Fletcher, Allen, Ramadan; Shaqiri, Choupo-Moting; Jese
CHELSEA STARTING XI: Courtois; Azpilicueta, Christensen, Rudiger; Moses, Bakayoko, Kante, Alonso; Willian, Morata, Pedro
One ball from Azpilicueta beats all three defenders to send Morata clean through on goal, and the summer signing makes no mistake with a cool finish into the bottom corner.
The visitors still have that all-important lead, though.
6 - Only Diego Costa (8) had more goals/assists in his first six PL games for Chelsea than Alvaro Morata (4 goals, 2 assists). Replacement. pic.twitter.com/p2wICSccL2— OptaJoe (@OptaJoe) September 23, 2017
The mistake comes from captain Fletcher of all people, who tries to chest the ball down to his defenders but only picks out Pedro. From there the Spaniard reaches the edge of the box before arrowing his finish into the bottom corner with unerring accuracy.
27 - Since 2016-17, only Diego Costa (29) has been involved in more goals in all comps for @ChelseaFC than Pedro (15 goals 12 assists). Key— OptaJoe (@OptaJoe) September 23, 2017
Jese may have been in a better position to go for goal from that cross, but he didn't shout loud enough for Diouf to hear him.
The balance of play itself has been very even, with Stoke being the better side if anything, but the champions have been clinical - capitalising on two lapses in the Stoke defence - and that has proved to be the difference so far.
From that point on it never looked in doubt that the Spaniard would finish it off, tucking his effort into the bottom corner for his first Premier League goal with his feet.
Again it was clinical from the champions as Fletcher's mistake gifted possession to Pedro, who immediately pounced on the loose ball before firing his finish into the bottom corner.
Stoke manager Mark Hughes will be fairly content with his team's all-round performance in that opening 45 minutes, but they need to offer more in the final third if they are to get anything out of this game.
Elsewhere, Everton's season looks to be going from bad to worse as they have just fallen behind at home to Bournemouth.
It is the correct decision from the referee, but Alonso should not have given him the decision to make.
It is another error which costs Stoke as Johnson gives the ball away to Morata, who is suddenly left against just Fletcher in the Stoke half. The captain cannot live with Morata's pace as he takes him on down the channel before opening his body at the last second and beating Butland with a superb finish from a tight angle.
Hazard flicks the ball to Fabregas, who spots the run of Azpilicueta and picks it out with a lovely scooped pass into the area. From there, Azpilicueta coolly chests the ball into the path of Morata, who is left with a simple finish past Butland to give this scoreline an even more one-sided look.
The Potters played well for the most part - better than Chelsea, in fact - but the Blues were so clinical in front of goal and their first three strikes came as a result of Stoke errors. After that they did begin to dominate, but the final scoreline is very harsh on Stoke.