Podolski's goals were set up by fellow Germans Mezut Ozil and Per Mertesacker as the 10-times winners eased into the last 16 on a rainy night in north London.
The Polish-born forward rifled home from inside the area after a pass from midfielder Ozil in the 15th minute and headed in a corner at the far post from defender Mertesacker's flick-on at the near post in the 27th.
Podolski was replaced by Olivier Giroud in the 79th and within five minutes the Frenchman had scored Arsenal's third goal with a tap-in from left back Kieran Gibbs' cross.
Fellow substitute Santi Cazorla completed the rout in the 89th from a rebound after keeper Joe Murphy had parried a shot.
Three minutes after the opening goal Coventry, the 1987 FA Cup winners who now play in League One, almost hit back with Arsenal stopper Lukasz Fabianski making a diving save to keep out midfielder Carl Baker's fine shot from outside the area.
Striker Leon Clarke also missed two good chances to get a more enterprising Coventry side back into the game early in the second half.
Fabianski saved his first effort and the striker then hit the post from a central position inside the box after Franck Moussa's square pass with the second.
"In the first half we did the job and took the game to them in a serious way and played the way we know we can," Arsenal manager Arsene Wenger was quoted as saying by the BBC.
"In the second half we dropped off and, give them credit, they gave it a go. Coventry should be pleased with their performance and we should be pleased with the result."
Former European champions Nottingham Forest, now in the second tier Championship, were held 0-0 at home by League One side Preston North End in a clash of former FA Cup winners.
The fourth round continues on Saturday with holders Wigan Athletic, who were relegated from the top flight three days after their dramatic late win over Manchester City in last season's final, at home to Premier League Crystal Palace.
Cup favourites Manchester City host second-tier Watford, also on Saturday, while seven-times winners Chelsea face Stoke City in an all-top flight tie at Stamford Bridge on Sunday.
(Writing by Rex Gowar; Editing by Ken Ferris)
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