The German forward put the hosts ahead in the 15th minute from a Mesut Ozil pass and doubled the advantage 12 minutes later after fellow compatriot Per Mertesacker headed on a corner.
Coventry improved vastly after the break and Leon Clarke hit the post but Arsenal substitute Olivier Giroud made sure of the win by adding a third with six minutes remaining after a Kieran Gibbs pull back.
Fellow substitute Santi Carzorla put gloss on the result from close range with a minute remaining after Joe Murphy could only parry Carl Jenkinson's angled shot into his path.
Gunners boss Arsene Wenger had named a strong starting XI, which included club-record signing Ozil.
Wenger, who also handed a senior debut to 16-year-old midfield prospect Gedion Zelalem - not even born when he first took charge at Highbury - will now regroup his squad for the Premier League trip to Southampton on Tuesday having cruised through without really getting out of second gear.
Arsenal showed their intentions from kick-off, as Jack Wilshere and Podolski both went close before the Gunners swept into the lead on 15 minutes when the German forward rounded the goalkeeper to score from a tight angle.
Coventry captain Carl Baker forced a good save from Arsenal number two Lukasz Fabianski before Podolski headed in a second at the far post from a corner.
There was a protest from the 5,000-strong travelling support on 31 minutes, with placards held up asking 'Why?' the Midlands club had been forced to move 'home' matches 31 miles down the M1 to Northampton, with the Arsenal supporters giving their own applause to show solidarity as chants of "we want to go home, back to the Ricoh" rang out in the Clock End.
Podolski was inches away from converting his hat-trick as he collected Jenkinson's cross and turned to fire the ball just over.
There was some concern after Wilshere picked up a knock to his ankle in a 50/50 with Cyrus Christie, but the England midfielder was able to continue.
As half-time approached, some of the floodlights went out, which led to thousands of smart phones being waved around the stadium.
The lights were on again for the restart and Coventry had a great chance to get back into the tie as Clarke was sent clear into the right side of the Arsenal box.
However, the want-away forward - linked with a switch to League One rivals Wolves - saw his angled shot blocked at the near post.
At the other end, Alex Oxlade-Chamberlain felt he should have been given a penalty when pushed over by Christie, but referee Robert Madley was unimpressed.
Clarke then hit the outside of the post when he picked out on the edge of the Arsenal penalty area by Franck Moussa.
On 61 minutes, another protest from the Coventry supporters followed, this time with "When?" placards, while singing the 'Sky Blues Song' which was written by Jimmy Hill, who became their manager in 1961.
Bizarrely no stewards move to intercept the Coventry fan who then came on to the pitch, and it took the intervention of a couple of Sky Blues players to help usher him off.
With 20 minutes left, Denmark forward Nicklas Bendtner, fit again from an ankle problem, was replaced by Cazorla and Oxlade-Chamberlain came off as German-Ethiopian midfielder Zelalem - who spent some time growing up in the United States and turns 17 on Sunday - made his senior Arsenal bow.
Giroud was sent on for the closing stages and needed no second invitation to sweep home a low cross from Gibbs as he made it 3-0 on 84 minutes.
Cazorla completed the win after Jenkinson's angled shot was parried by Murphy before Ozil hit the post as the Gunners finished strongly.