The hosts were dominant during the first half but could not find a way past Doncaster goalkeeper Sam Johnstone.
However, Vokes tucked away a penalty two minutes after the break and Stanislas doubled Burnley's lead with quarter of an hour to go to hand Rovers a first defeat in five.
Sean Dyche named an unchanged team to the one that beat Charlton 3-0 at the weekend and his Burnley side dominated the first half.
In the 10th minute Scott Arfield grazed the top of the crossbar with a curling effort from the edge of the area, before David Jones and Dean Marney had efforts blocked on the edge of the area.
Stanislas was the first to test Johnstone when his cross-shot was palmed away by the Doncaster keeper as the Clarets continued to threaten the visiting defence.
Paul Dickov's side did not muster many chances of their own, but Chris Brown came close to taking the lead with a header that drifted narrowly wide.
Stanislas then had two chances to take the lead, firing over the bar before forcing Johnstone to palm away a dipping effort.
Burnley continued to look for the opening goal and were unlucky when Arfield could not convert a rebound after Johnstone had spilled Jones' swirling effort.
Arfield and Vokes both dragged efforts wide before Johnstone made a string of saves to keep the scores level at half-time, keeping out first Arfield and then a powerful Marney drive.
Moments into the second half, Burnley were given a chance to take the lead from the penalty spot after the referee ruled that James Coppinger had pushed Vokes. The striker then stepped up and placed his penalty into the top corner for his 20th league goal of the season.
Vokes thought he had doubled the Clarets' advantage soon after but his strike was ruled out for offside.
Clearly not content with their one-goal lead, second-placed Burnley came close to adding another when Stanislas whipped a free-kick narrowly wide.
Billy Sharp then had a rare opportunity to bring Doncaster level, but the on-loan Southampton striker volleyed over the bar.
Barnes should have scored for Burnley in the 72nd minute but, after his first effort was blocked, the striker lashed horribly over.
However, Burnley did grab their second when Stanislas calmly slotted past Johnstone after Jones had played the winger through on goal.
Rovers threatened to find a way back into the game when David Cotterill's free-kick hit the bar and Abdoulaye Meite hammered over.
Tom Heaton was then forced into action for the first time in the game in the 85th minute, getting down low to keep out a Brown effort, with Burnley easing to victory.