Tubbs pounced 10 minutes from time to cancel out Nathan Byrne's opener as the Reds extended their unbeaten run to five games.
Swindon are now without a win in four matches as their charge for the top six suffered another blow.
Crawley were unlucky not to be awarded a penalty 10 minutes before the break when Jamie Proctor was grounded by Darren Ward.
Dany N'Guessan should have opened the scoring on the stroke of half-time, but his tame effort from a few yards out was straight at Paul Jones.
It was Swindon who went ahead three minutes into the second half as Byrne flicked in Jack Barthram's cross at the near post.
The visitors drew level when Tubbs turned home Josh Simpson's low cross in a half of few chances.