The 29-year-old was imperious throughout the Championship fixture and helped the home side end a run of four consecutive defeats, during which they conceded 13 goals.
After edging the encounter, Coppinger provided the decisive 59th-minute cross that Burnley's Michael Duff could only divert into his own goal.
The Northern Irishman's mishap ended the Clarets' five-match unbeaten run and saw Eddie Howe taste defeat for the first time as manager of the Lancashire outfit.
Doncaster were boosted heading into the clash by the inclusion of skipper Brian Stock after a calf injury, while Sam Hird deputised in central defence for the injured Matt Kilgallon.
Rovers boss Sean O'Driscoll also handed 20-year-old goalkeeper Gary Woods his first ever start after first-choice Neil Sullivan picked up an injury.
Burnley boss Howe, meanwhile, dropped Steven Thompson from the visitors' attack in favour of new signing Charlie Austin, who joined from Swindon on Friday.
The inclusion of Austin was the only change to the side that knocked Burton out of the FA Cup at the weekend - and his inclusion almost paid dividend within four minutes.
Jack Cork sent a drilled cross in from the right wing that Woods could only parry to Austin, who should have done better in the six-yard box.
Aside from Austin's opportunity, chances were sparse in the opening exchanges as the two sides cancelled one another out.
Rovers began to exert an increasing amount of pressure midway through the first half, with Coppinger seeing a shot deflect into the side netting before James O'Connor flashed a shot wide from the resulting corner.
James Hayter was unfortunate to put wide in the 29th minute when connecting with a cross at the back post as the home side surged forward in droves, looking particularly dangerous down the flanks.
Hayter was proving a menace in attack, firing inches wide before Danny Fox's goal-line clearance prevented the former Bournemouth man breaking the deadlock on the stroke of half-time.
Coppinger was the vital component in both moves and proved integral to Rovers once more after the break.
Having been clattered by Danny Fox moments earlier, Coppinger attacked the now yellow-carded full back and whipped in a dangerous-looking cross that Duff managed to header into his own goal.
Burnley, seemingly spurred on by the goal, pressed forward in numbers, coming close through Dean Marney's low drive and Jay Rodriguez's looping header.
Chris Iwelumo was agonisingly close to connecting with a free header out with 10 minutes remaining as the visitors piled forward with intent.
O'Driscoll's side, however, were able to grind out the victory and Burnley's defeat was compounded by a worrying-looking thigh injury sustained by substitute Martin Patterson.