Jason Scotland gave the Swans an interval lead, but after Ferrie Bodde was dismissed in the 62nd minute, Wellens struck twice to give his side their first league win of the season.
It was the visitors who had the first strike on goal when Kevin Austin was dispossessed by Mark Wilson in the tenth minute, but the former Manchester United midfielder dragged his shot wide.
Brian Stock and Wilson controlled midfield, keeping the Swans pinned back and Gareth Roberts let fly with a drive that zipped beyond the post with Dorus de Vries rooted to the spot.
The Swans went ahead against the run of play in the 40th minute. Donny keeper Neil Sullivan failed to hold on to an Andy Robinson cross and Scotland was unmarked as he turned the ball home from close range.
Five minutes into the second half, Sullivan had to leap to his left to push behind a Bodde shot that looked a goal all the way.
That was the last action for the Dutchman as he was sent off following an off-the-ball skirmish with Stock.
Rovers suddenly found a head of steam to equalise in the 68th minute with a superb strike from Wellens after a clever pass from the busy Stock.
Both sides made use of their substitutes in the closing stages, but Rovers were now on top and Wellens struck his second in the 83rd minute with a 25-yard volley to secure all three points for his side.