The decisive moment came on 26 minutes when Oscar fired a spectacular free kick past the helpless Asmir Begovic.
The home side had two good chances to kill the game off after that, but Oscar and Andre Schurrle both hit the woodwork.
Here, Sports Mole takes a closer look at the action between the teams in West London.
On target: 5
On target: 1
Was the result fair?
Chelsea made life difficult for themselves as they struggled to put Stoke to bed, but they were ultimately deserving of their place in the fifth round thanks to a commanding display throughout the 90 minutes at Stamford Bridge.
Jose Mourinho's side will be happy with their progress in the competition, but they made hard work of it come the final whistle. The Blues looked good going forward as the passing and movement of their attacking players continually caused Stoke problems. Meanwhile, they were solid enough in defence to keep the visitors at bay for much of the match. However, there will be a nagging worry at Stamford Bridge that their goalscoring troubles from earlier in the season are still lying dormant. Their inability to score a second goal allowed Stoke the opportunity to get something from the game. Luckily for the hosts, the Potters did not.
The Potters offered very little and were rightly knocked out of the competition. They defended doggedly enough, but they failed to impress in midfield and attack. As the scoreline stayed the same, one would have expected them to push more men forward in search of an equaliser. However, they seemed content to be beaten instead of going down fighting. While manager Mark Hughes is sure to be pleased to be able to start focusing fully on the Premier League, one would have thought that his team could have shown a bit more determination and spirit.
Sports Mole's man of the match
Oscar: Not only did the Brazilian score the winning goal with a dazzling free kick, but he also provided plenty of dynamism for Chelsea moving forward. His flair and panache made sure that Stoke were never able to relax as he linked up well with his teammates in the final third.
Chelsea had a great chance to extend their lead during the second half, but Samuel Eto'o somehow poked the ball wide from close range when it looked easier to score.
Chris Foy turned down a couple of contentious penalty shouts for Chelsea which incensed the home fans, but the official got them both right during what was a solid display.
Chelsea: The Blues are back in Premier League action on Wednesday when they welcome London rivals West Ham United to Stamford Bridge.
Stoke: The Potters continue their Premier League campaign on Wednesday by making the trip to Sunderland.