They started well by winning a corner in the first minute but they should have taken the lead soon after when a through ball from Angel Rangel left Shefki Kuqi free on goal but he hurried his shot and it flew high over the bar.
It was all Swansea as they looked for the opening goal to settle nerves but Kuqi squandered more good opportunities and midway through the half Darren Pratley lofted a shot high and wide.
Donny came close just after the half hour with their first attempt on goal when a free-kick from Jay Emmanuel-Thomas dipped late to just clear the home cross bar
Back came the Swans and a shot from Pratley was deflected inches wide with Donny keeper Ben Smith beaten before minutes later he fired over after Kuqi had made the opening.
The half ended with the Swans well on top but still looking for that elusive goal.
Once again it was Pratley who ran through in the opening minutes of the second half to meet a pass from Nathan Dyer but, under pressure, he sent his shot wide of the far post.
In the 52nd minute the crowd were on their feet as David Cotterill volleyed goalwards but the ball struck the side netting.
Gorka Pintado replaced Mark Gower just before the hour as the Swans went for a two man attack but Kuqi left the field a little later with Craig Beattie coming on to partner Pintado up front.
Pratley picked out Beattie down the left but the striker delayed his shot and it was blocked by James O'Connor as the Swans swarmed forward.
With ten minutes remaining, Lee Trundle was introduced but it was Cotterill who went close when he flashed a drive narrowly wide a minute later.
Doncaster could have sewn up an unlikely victory when substitute Paul Heffernan was left unmarked in front of goal but he blasted high and wide.
With time running out there were loud appeals for a penalty as Sam Hird appeared to bring down Beattie inside the area but referee Grant Hegley waved play on.
Trundle did net in the last minute of time added on but it was disallowed for handball and the Swans play-off hopes were dashed.