Steven Gerrard halved Arsenal's advantage with a 59th-minute penalty after Podolski was adjudged to have tripped Luis Suarez.
The visitors felt they should have been awarded a second spot-kick moments later after Oxlade-Chamberlain appeared to foul Suarez, but their appeals were dismissed by referee Howard Webb.
Victory in a game splattered with yellow cards means that Arsenal will host Everton in the last eight, but they will have bruises to nurse ahead of Wednesday's meeting with Bayern Munich in the UEFA Champions League last 16.
Everton beat Swans to advance
Everton's reward for a fifth round FA Cup win over Swansea was a tough quarter-final draw away to Arsenal, but manager Roberto Martinez will be pleased with the showing of recent signing Lacina Traore.
The Toffees advanced into the last eight with a 3-1 win over another Premier League side in the Swans at Goodison Park.
Martinez, who won the Cup with Wigan last season, saw Lacina Traore, his loan transfer signing from Monaco in January, take just four minutes of his debut to give the Merseysiders the lead.
Jonathan de Guzman drew Swansea level in the 15th before second half substitute Steven Naismith put Everton in front again -- with his fourth goal in six games -- and then won the penalty from which Leighton Baines scored to put the result beyond doubt.
However, there was a worrying moment for Everton when, shortly before full-time, Naismith went off with concussion after being caught in the head by Jordi Amat.
"The draw hasn't been too kind in terms of playing away from home, but we will embrace it and look forward to it," Martinez told ITV Sport.
As for Naismith, he added: "We are a little bit concerned by his concussion."
Naismith came on for Traore and Martinez was pleased with the striker's display.
"It's important," he said. "Lacina has been working very well. We needed to introduce him slowly. It was pleasing to see him on the pitch and see him scoring."
Sheffield United keep dream alive
Sunday's other Cup tie saw third tier Sheffield United beat second-tier Nottingham Forest 3-1.
United, managed by former Forest favourite Nigel Clough, could be involved in a derby quarter-final against Sheffield Wednesday if their city rivals beat Charlton Athletic in a re-arranged match.
Jamie Paterson headed Forest into an early lead before the on-loan Conor Coady drew the Blades level in the 66th minute.
And Chris Porter then sparked scenes of near delirium amongst the Bramall Lane faithful with two goals right at the end of the match.
He put them in front with a 90th minute penalty following Greg Halford's handball and then sealed victory in stoppage-time with a finish from close range.
The last eight line-up also saw FA Cup holders Wigan paired against Manchester City in a repeat of last season's final.