The Brazilian had dislodged Mata from his position as Chelsea's main playmaker, prompting the Spaniard to join Manchester United, and the quality of his goal demonstrated why manager Jose Mourinho rates him so highly.
The Blues failed to add to Oscar's first-half effort despite hitting the woodwork twice, although it was a largely comfortable victory, with Stoke rarely threatening to score an equaliser and force a replay.
Chelsea's reward is a mouthwatering fifth round trip to English Premier League rivals Manchester City.
Mourinho opted to rest a number of senior players including John Terry and Petr Cech, but his decision to include Samuel Eto'o in his starting line-up raised eyebrows.
The forward scored a hat-trick in the 3-1 victory over Manchester United seven days previously, but the manager had been expected to rest Eto'o and hand Demba Ba a rare start.
However, Eto'o wasted little time justifying Mourinho's decision, quickly impressing himself on the Stoke back-line and drawing the foul that allowed Oscar to break the deadlock in spectacular style.
Eto'o's smart turn and shot in the second minute suggested this would be a long afternoon for Stoke, who were hoping to repeat their Premier League victory over the Blues in early December.
The visitors' initial response suggested otherwise, with Peter Crouch heading wide after getting ahead of his marker David Luiz three minutes later.
Stoke struggled to put Cech's replacement Mark Schwarzer under real pressure, but they matched Chelsea in the midfield areas for long periods.
Chelsea, though, looked capable of capitalising on any Stoke error and should have taken the lead from a breakaway after Marco Arnautovic gave the ball away carelessly inside the home half.
Then, when Erik Pieters was penalised for a tackle from behind on Eto'o, Oscar produced a memorable free-kick that gave Mourinho's side a 27th-minute lead.
Begovic was back in the side after a six-game absence with a broken finger, but no blame could be laid at his door.
Having finally broken through Stoke's defence, Chelsea took control of the remainder of the first half, although Stephen Ireland did provide a scare when he fired a left-foot shot into the side-netting.
Oscar should have added a second goal when he broke into Begovic's area and beat the goalkeeper with a powerful shot that cannoned to safety off the upright.
And Chelsea maintained their momentum after the break when Andre Schurrle found himself in a similar position before firing against the crossbar.
Stoke offered little in return and could have been on the end of a more comprehensive defeat if Eto'o had not missed from close range late on, while Ramires had a penalty appeal turned down.
Sheffield force Fulham replay
Fulham survived an FA Cup scare against third-tier Sheffield United on Sunday by battling back to secure a 1-1 draw at a mud-caked Bramall Lane in the fourth round.
Fulham made 11 changes to their starting line-up and League One strugglers United looked set for a repeat of their third-round win over Aston Villa after Chris Porter gave them a 31st-minute lead.
Heavy rain in northern England made for a muddy, sapping playing surface and the game only went ahead after a late pitch inspection.
There was an early blow for the visitors when John Arne Riise had to go off with an apparent hamstring injury and with half an hour played, United went ahead.
Young centre-back Harry Maguire made an authoritative burst into the Fulham box before driving in a low cross that was stabbed home by the opportunistic Porter.
In reply, Rodallega was denied a penalty after tumbling over a challenge from United goalkeeper George Long, while Pajtim Kasami shot narrowly wide early in the second half.
Doyle was shown a straight red card in the 52nd minute for kicking out at Chris David, but the hosts continued to threaten, with Ryan Flynn working Fulham goalkeeper David Stockdale and Jose Baxter firing wide.
Nigel Clough's side then had a very strong penalty claim turned down after Flynn was caught by a clumsy challenge from visiting captain Aaron Hughes.
Fulham threatened to complete the comeback with seven minutes remaining, but Philippe Senderos saw his glancing header bounce back off the crossbar.