All the goals came in the second half following a lacklustre performance by both teams before the interval but, when Rovers continued to struggle in the second period, the Boro raised their game and finished worthy winners.
Francois Zoko fired the League One side in front following a mistake by Paul Quinn in the 49th minute and Peter Hartley headed home unchallenged following a 65th-minute corner.
Harry Forrester pulled one back in the 72nd minute with a spectacular 25-yard volley but Rovers were caught pushing too many players forward when Darius Charles added Stevenage's third in the last minute of normal time.
Teenage right-back Liam Wakefield hammered home a second for Doncaster deep into stoppage time but it came far too late to make a difference to the outcome.
Rovers were without midfielder Richie Wellens, who has yet to sign a new contract and has been linked with QPR, and his influential presence was sorely missed.
Mark Duffy landed the first shot on target in the 11th minute but visiting goalkeeper Chris Day reacted smartly to save.
Then Chris Brown failed to connect properly after latching on to a right-wing cross from Wakefield and his header sailed harmlessly wide.
Stevenage grew in confidence as Rovers failed to assert themselves and enjoyed a spell on top just before the interval.
Filipe Morais sent over a deep cross from the right which was only half cleared and Luke Freeman hit a shot on the turn which was saved at the near post.
Marcus Haber had a header deflected wide from a corner and then Freeman was out of luck again with a scorching 20-yarder which fizzed inches past the post.
Doncaster went from bad to worse in the second half and Stevenage cashed in when they snatched the lead in the 49th minute following a blunder by centre back Quinn.
The defender fumbled an attempted clearance and made a present of the ball to Zoko who took a few strides forward before slotting home in fine style.
Ross Turnbull came to the rescue to prevent Rovers going further behind when he did well to save a close-range effort from Freeman in the 56th minute.
Quinn partially redeemed himself five minutes later with a superb last-ditch tackle to prevent Zoko scoring again.
But there was no stopping Hartley when he headed home and after Forrester's scorcher, Charles put the issue beyond doubt from Zoko's right-wing cross.
Wakefield scored Rovers' second but Stevenage stayed resolute in defence and held on for the win.