The Brazil midfielder scored in the 27th minute at Stamford Bridge with a curling 20-meter shot that ended in the top corner as Stoke goalkeeper Asmir Begovic dived in vain.
Dangerous on the break and dominant in midfield, Chelsea almost doubled its lead soon after when Oscar hit the post. Andre Schurrle also hit the woodwork after the interval with a shot from a tight angle as Chelsea kept creating the better chances.
Chelsea manager Jose Mourinho, whose team will face Premier League title rival Manchester City in the next round, was happy to avoid a replay — although he said his players could have won by a bigger margin on his 51st birthday.
"I think 1-0 is short for the way we played and for the way we controlled the game. But the important thing today was to win and not play a replay, because we don't need an extra match and I think that was a good thing," Mourinho said.
Samuel Eto'o missed the target from close range after Ramires failed to convert Eden Hazard's precise cross, but the home side had done just enough by the final whistle.
Hovering four points above the relegation zone, Stoke has scored just 22 league goals this season but manager Mark Hughes is confident he will soon add Cardiff striker Peter Odemwingie to his attacking options.
"It should go through tomorrow ... we're pleased to get Peter in," Hughes said. "He's a good Premiership player with experience at this level, a goal in him and a bit of pace, which we need to add to the squad."
Hugo Rodallega scored Fulham's equalizer with a low shot that spared his side more misery this season. The London club is just one point above the relegation zone in the Premier League standings.
Chris Porter put the hosts ahead in the 31st minute as he diverted a shot past David Stockdale. But the League One club had Michael Doyle sent off for a kick at Muamer Tankovic early in the second half and could not hold on to its lead.
Sheffield United manager Nigel Clough praised his players for their grit after the 1-1 draw but said the referee's decision to send off his player was a bit harsh.
"There was some sort of contact, whether it was enough to warrant a red card I'm not sure," he said.
Fulham manager Rene Meulensteen, whose priority is to keep his team in the Premier League next season, made 11 changes for the tie against a side that knocked out Aston Villa in the previous round but is facing a tough relegation battle in its division.
On a waterlogged pitch, Sheffield United looked like it could hold on even after Doyle's sending off, and even threatened to extend its lead when Ryan Flynn led a swift counterattack down the right only to see his low shot stopped by Stockdale.
Both sides were denied penalties as Rodallega and Flynn appeared to be fouled in the area.
Meulensteen admitted his side lacked the quality to prevail but was happy with his youngsters' performance.
"They gave it their all and got their noses in front," Meulensteen said. "If there's any positives to take from the game there's two things. Obviously we're still in it, that's one, but the young boys we brought in did excellent."