Morecambe struck first as Bentley headed home Kieron Walmsley's far-post corner in the 12th minute.
The lead lasted until the half-hour mark, before Green picked up a Nicky Barmby pass and curled an excellent 20-yard shot past Adam Sollitt in the visitors' goal.
Danny Allsopp almost headed in Marc Joseph's cross to give Hull a half-time lead, but on the hour mark a half-empty KC Stadium fell silent in the main as Morecambe again forged ahead.
Garry Thompson was the provider as his ball in from the left allowed Twiss to fire past Boaz Myhill from close range.
Just six minutes later Keane made it 2-2 as he fired past Sollitt from just outside the area following good work by substitute Walters.
Walters then made the crucial breakthrough four minutes from time, converting Stuart Elliott's pass to break Morecambe hearts, although Danny Carlton almost set up a replay with an injury-time header.