Palace striker Easter pounced on a rebound in the fifth minute of time added on to snatch his side an unlikely three points after Sean Scannell had equalised for the home side in the 90th minute with his second goal of the season.
Lukas Jutkiewicz had put Coventry ahead three minutes into the second half after a dour opening 45 minutes, but Palace were rewarded for their never-say-die approach when it appeared as though the visitors had clung on for their first win of the season.
South Africa international Kagisho Dikgacoi replaced Darren Ambrose in the Palace midfield and Coventry also made one change, with Conor Thomas in for midfielder David Bell.
Neither side had a shot on target before Coventry were dealt a blow in the 18th minute when striker Gary McSheffrey was forced off injured and replaced by Joshua Ruffels.
Palace striker Scannell forced Joe Murphy into the first save of note, the Coventry goalkeeper launching himself full length to keep out a fierce 20-yard drive, while at the other end visiting midfielder Gael Bigirimana, a Burundi-born youth team product at the Ricoh Arena, fired an ambitious 35-yard effort over the crossbar.
Scannell worked another opening with a clever dummy inside the penalty area allowing him to unleash a shot on target, which was blocked by Coventry defender Martin Cranie.
Palace had their own injury problems to contend with in the 39th minute when the Dikgacoi was withdrawn from the action and replaced by Wilfried Zaha as the first half ended goalless.
Jutkiewicz broke the deadlock three minutes into the second half. Defender Cyrus Christie crossed from the right and after the ball ricocheted off Palace defender Peter Ramage, the Pole turned the ball home from six yards.
Scannell fired in another effort that Murphy did well to hold on the hour as Palace responded and three minutes later substitute Zaha dragged a shot from the edge of the area wide.
Roy O'Donovan and Paddy McCarthy saw efforts fly over the crossbar for Palace, but Jutkiewicz went close to adding a second with an angled shot from inside the area that was beaten away by Palace goalkeeper Julian Speroni.
Palace striker Easter was next to spurn a golden chance for the home side when he headed wide in the 82nd minute before Jutkiewicz was again denied another goal by Speroni, who tipped his snap shot over the crossbar.
But the home side continued to pour forward in search of the equaliser in the closing stages and Zaha was just off target with a thumping drive.
Then Palace's luck changed in the final minute thanks to inspiration from Scannell.
Zaha's goalbound shot from the edge of the area was kept out by Murphy, but Scannell pounced on to the bouncing rebound to send a spectacular scissor kick into the bottom corner from 12 yards.
Then in the fifth minute of time added on Easter jumped on to another parry from Murphy, again prompted by a Zaha effort, and converted from close range to complete an unlikely turnaround.
Crystal Palace 2-1 Coventry: Match Report - view commentary, squad, and statictics of the game as it happened.