That sparked a vibrant response from the visitors, with Steven Gerrard halving the deficit from the spot after Luis Surez had been felled by Podolski. Łukasz Fabiański then did expertly to deny Daniel Sturridge one-on-one, and Daniel Agger headed wide as Arsenal rode out the storm.
At Goodison Park, meanwhile, the tie was four minutes old when Lacina Traor marked his Everton debut with a goal. Jonathan De Guzmn quickly restored parity with a header, but Everton wrestled back the initiative after the interval, Steven Naismith restoring their lead before a Leighton Baines penalty completing the formalities.
As for third-tier Sheffield United FC, they posted a 3-1 comeback victory against Nottingham Forest FC at Bramall Lane to set up a possible Steel City derby in the next round, Chris Porter registering twice in added time.