Tubbs raced onto a fine throughball 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.
Crawley, currently second in their first league season, oozed confidence from the start with Tubbs sending Mannone scrambling early, and Sanchez Watt - signed from Arsenal until the end of the season - causing problems with his pace.
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.
With the underdogs' attacks unrelenting, Tyrone Barnett beat Mannone on the by-line but drilled his attempted cross at the legs of Daniel East. In a rare response from Hull, Liam Cooper headed well wide.
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.
Nick Barmby's men finally managed their first shot on target in the 39th minute when Tom Cairney unleashed a low, long-range effort from the left which drifted into the hands of Crawley keeper Scott Shearer.
And Robbie Brady came closest one minute before the break when he lashed another long-range strike which took a deflection in the box and was well saved by Shearer diving to his left.
Hull's revival continued just after break when 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 cutting in from the left flank, 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.
Shearer was fortunate to escape in the 68th minute when he raced from his box to reach a through-ball and was beaten to it by Hull substitute Aaron Mclean, who clattered his attempted lob against the keeper.
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.