Tubbs raced onto a fine through-ball from Sergio Torres to slide the only goal of the game under Tigers keeper Vito Mannone and revive dreams of last year's last 16 trip to Old Trafford.
Steve Evans' underdogs richly deserved their win over a Hull side featuring seven changes to their last league outing, with on-loan new boy Sanchez Watt agonisingly close to extending their margin of victory late on.
Tubbs aimed an overhead kick just over Mannone's bar after a lofted pass by Torres, and a Watt shot put more pressure on the home keeper as Crawley completely dominated the opening quarter of an hour.
Crawley created another fine chance on 32 minutes when Tubbs was denied the chance to get on the end of Barnett's cross in front of goal by a superb interception from Joe Dudgeon.
Robbie Brady came closest for Hull one minute before the break when he lashed another long-range strike which took a deflection in the box and was well saved by Scott Shearer diving to his left.
Hull's revival continued just after break when Tom Cairney saw a curling effort flapped behind by Shearer, but the visitors came close at the other end when captain Pablo Mills drilled a shot which Mannone did well to tip wide.
Crawley finally broke the deadlock in the 57th minute when Torres sent Tubbs clear and the striker kept his poise to squeeze the opener home under Mannone. The goal sparked the game into life, with Richard Garcia almost prodding Hull level within two minutes only for a full-stretch Shearer to scoop the ball off his goal-line in a crowded box.
Crawley should have extended their lead in the 75th minute when Watt brought a point-blank save out of Mannone and from the resulting corner the same player somehow hit the outside of the post with the goal at his mercy.
Hull's last chance to salvage an equaliser fell to substitute Mark Cullen in injury time but his swinging shot was blocked in a crowded box and Crawley hung on to secure another famous win.