The striker capitalised on poor defending to lift the ball over advancing 'keeper Michael Poke after 15 minutes of the League Two clash, and Matt Lund's free-kick doubled the advantage just two minutes later before the visitors had full-back Joe Oastler dismissed for a second booking on the half hour.
Back roared Torquay, though, and Rene Howe pulled one back with a curling effort after the break before Aaron Downes' header levelled matters shortly after. But Eaves netted the winner from substitute Wayne Brown's 76th-minute pass.
A cracking West country derby saw Rovers dominate the first half before their opponents emerged to give a much better account of themselves after the interval.
Both sides had chances in a feast of attacking football, but the home side ran out worthy winners and have now been beaten only once in seven games after a poor start to the season.