Eastwood's smash-and-grab goal midway through the second half condemned the Pilgrims to a second successive home defeat - and sent them to the bottom of the Championship.
A dour first-half struggle saw neither side take control and chances were few and far between.
For Argyle, Jamie Mackie gave Coventry left-back Patrick van Aanholt a few problems and created a chance for himself when he rounded the defender and cut back from the byline before firing off a shot that Leon Barnett blocked.
A clearly confident Eastwood was the visitors' main danger man and it took a fine last-ditch tackle from Shane Lowry to prevent Michael McIndoe from converting the Wales international's low cross.
Argyle survived two scares of their own making before the interval.
First defenders Reda Johnson and Kari Arnason got in each other's way to allow Eastwood a clear sight of goal, but the Coventry man took the chance too early and shot wide.
Then in first-half injury-time, goalkeeper Romain Larrieu spilled an Eastwood shot and the ball eventually broke to Barnett, who fired the ball over an unguarded goal from five yards.
Argyle captain Carl Fletcher had a shot from the edge of the penalty area blocked by Coventry goalkeeper Keiren Westwood in the opening stages of the second half.
An Alan Judge cross was headed over by Rory Fallon, who then went even closer when he connected with Lowry's delivery from the opposite flank.
However, Eastwood relieved the pressure with a picture-book goal, heading home David Bell's perfect cross and, strive as they did, the Pilgrims could not claw anything back.
Fallon saw a shot deflected narrowly wide and then Arnason saw a header blocked on the line by Bell before Westwood saved from the same player in injury time.