Craig Mackail-Smith scored a quickfire double midway through the second period but Eastwood's third made it three goals in six minutes to seal the win for the hosts.
Coventry's first chance of any note came after 17 minutes when a long Stephen Wright throw found Eastwood, but his effort from the edge of the area was tame and Joe Lewis collected the ball with ease.
The visitors broke quickly but George Boyd's fierce shot was palmed over the bar by Keiren Westwood.
The game started to become very scrappy before Coventry took the lead after 34 minutes.
A David Bell corner was flicked on by Martin Cranie and Eastwood was unmarked at the back post to tap the ball home.
A minute later Eastwood added his second of the game. Aron Gunnarsson's long throw was flicked on by Leon Barnett and the former Wolves striker headed home.
Gunnarsson then tested Lewis with an effort from the edge of the area before a superb snap-shot from Leon Best was clutched at the second attempt by the Posh goalkeeper.
The visitors pushed forward early in the second half and Mclean could have pulled a goal back but fluffed his lines six yards out and sliced the ball wide.
Lee Frecklington then delivered an inch-perfect corner which was headed over by Craig Morgan as the visiting fans started to get behind their side.
The visitors were given a lifeline with 24 minutes left when a long throw from Exodus Geohaghon caused confusion in the Coventry area and Mackail-Smith was on hand to slot the ball home.
In similar fashion to Peterborough's capitulation after Coventry's opener, the hosts themselves fell apart following Mackail-Smith's strike and within five minutes the scores were level.
Boyd unleashed a low strike at goal which Westwood was unable to hold and Mackail-Smith followed up to complete a simple finish from 12 yards.
The scores remained level for all of 60 seconds before Eastwood completed his hat-trick.
A foul by Frecklington on Gunnarsson saw McIndoe deliver a long ball deep into the Posh penalty area and Eastwood was there to smash the ball into the roof of the net.
Frecklington had the best chance to equalise in the closing stages but his long-range effort curled wide of the upright as Mark Cooper continues to search for his first win since taking charge of Peterborough.