In a game that Arsenal dominated from the start, Podolski put the hosts ahead in the 15th minute by slotting home a cool finish after being set up by Mesut Ozil. He then doubled the lead with a close-range header from a corner in the 27th.
The win was never in doubt after that and Arsenal added two late goals as substitute Olivier Giroud struck in the 85th after replacing Podolski five minutes earlier, before Santi Cazorla slotted in the fourth.
In Friday's only other FA Cup game, Nottingham Forest was held to a 0-0 draw at home against Preston.
Playing in a steady drizzle in north London, it was the kind of comfortable victory Arsenal had hoped for as it looks ahead to facing Bayern Munich in the Champions League knockout round, while trying to protect its slim lead atop the Premier League.
Despite playing at home against third-tier opposition, manager Arsene Wenger named a surprisingly strong starting lineup with both Mesut Ozil and Jack Wilshere in midfield — perhaps worried about another upset after going out to Blackburn last season.
And with Wenger contemplating buying a striker in the January transfer window, Podolski showed off his scoring touch by deciding the game in the first half.
Podolski was first sent clear by Ozil on the left side of the area and calmly rounded goalkeeper Joe Murphy before slotting a left-footed shot into an empty net.
The second came from a corner that Per Mertesacker met at the near post and headed on across the goal, with Podolski waiting at the far post to nod it into the net.
"Podolski can score goals and when you have a goal chance you want him to have it," Wenger said. "He is very accurate in his finishing."
Coventry was more enterprising after the break, enjoying several spells of pressure, but Leon Clarke wasted the team's two best chances for a goal.
Clarke raced clear down the right flank in the 50th but got too close to the byline before firing a low shot from a tight angle that Fabianski parried.
He then hit the right post with a first time effort from just inside the area as Arsenal seemed content to sit back and protect its lead.
"I know it was 4-0 but when you consider the difference in resources of both clubs, I know it's a hard luck story, but my players should take so much confidence from that performance," Coventry manager Steven Pressley said.
Giroud took his only chance in front of goal by slotting in a finish inside the near post and Cazorla continued his scoring streak from a similar position for his third goal in two games.
Arsenal's talented 16-year-old midfielder Gedion Zelalem had also come on as a substitute, as Wenger for the first time gave a debut to a player who was born after the Frenchman became manager of the team.
"That is more damaging for me than for him," Wenger joked.
Zelalem has been touted as perhaps the club's biggest talent since Cesc Fabregas and was given a rousing reception when he came on.
Coventry is a former Premier League club and the 1987 FA Cup winner but has struggled both on and off the field in recent years. The team is 11th in League One this season and has been forced to play all of its home games at Northampton because of a rent dispute with the owners of Ricoh Arena.
The contingent of about 6,000 Coventry fans protested the situation during the game, holding up signs in the 35th and 61st minute.
Arsenal had its own stadium problems, with some of the floodlights at the Emirates going out just before halftime. The game continued without interruption, with hundreds of fans making fun of the situation by lighting up their mobile phones in the stands in mock efforts to illuminate the pitch.
It was about the only thing that went wrong for Arsenal on the night.