Eastwood hit two first-half goals which were cancelled out by Craig Mackail-Smith in the second half, but the ex-Wolves man fired his third in the 72nd minute to claim the win for City.
It was a nervous start as both sides looked to end awful runs.
Coventry looked particularly anxious after going ten games without a victory and Peterborough had a couple of half chances early on.
Aaron McLean scooped a shot over the bar after seven minutes before Lee Frecklington cracked over the top from the edge of the area after a neat pass from Chris Whelpdale.
Keiren Westwood, the Coventry goalkeeper, needed a strong hand to keep out George Boyd's low shot after 20 minutes and then clutched Frecklington's shot out of the air.
Mackail-Smith could have opened the scoring after 24 minutes but headed over the bar as he was allowed a free header in the box.
The home team finally started to get into their stride after the half-hour mark as Aron Gunnarsson's long throw started to become a problem for Peterborough.
But it was the Icelandic international's head that led to the opening goal after 33 minutes as he knocked down David Bell's corner and Eastwood was on hand to score from close range.
It got even better for Coventry two minutes later as Leon Barnett flicked on Gunnarsson's throw and this time Eastwood headed in from inside the six-yard box.
City suddenly looked like a different team thanks to the two goals and both Gunnarsson and Leon Best went close to extending the lead before the break.
Peterborough should have reduced the deficit just after half-time when Westwood failed to get enough of a hand on Exodus Geohaghon's long throw and it dropped kindly for McLean. The striker snatched at the chance and fired over the bar from six yards.
Posh did get one back in the 66th minute when another Geohaghon missile dropped for Mackail-Smith who scored from close range.
The visitors drew level four minutes later as Boyd's shot was palmed away by Westwood and Mackail-Smith was in the right place at the right time to score.
Coventry had to put their foot back on the gas and they regained the lead in the 72nd minute when Michael McIndoe's free-kick was knocked on by Best and Eastwood drove the ball into the roof of the net.