The Dutchman was dominant on his first serve while racing to a 6-2 5-1 lead against Serbia's Tipsarevic but the former US Open quarter-finalist would rally to survive four match points and force another game.
Haase powered through to win, however, and he will face Marcel Granollers in the last eight. The Spaniard held his nerve to beat surprise German Open finalist Federico Delbonis, the Argentinian slumping to a 6-2 7-5 defeat.
Delbonis had eliminated Roger Federer on his way to the last two in Hamburg last week and the Swiss master is a potential opponent for Victor Hanescu after Roberto Bautista Agut retired hurt.
Romanian Hanescu, who was leading eighth-seeded Bautista Agut 7-6 (7/2) 3-2 when the Spaniard failed to recover from a medical time-out, will take on the winner of Federer's clash with big-hitting German Daniel Brands.