Ulster's Matt McCullough (left) holds up Thom Evans
Glasgow: (0) 8
Ulster: (7) 19
Tries: R Best, N Best, Bowe
Cons: Wallace 2
Glasgow Warriors' run of six home wins was ended by Magners League leaders Ulster at Hughenden.
Early Glasgow pressure came to nothing and Rory Best dived over to put Ulster ahead, with Paddy Wallace converting.
Dan Parks kicked a penalty after the break for Glasgow, but Neil Best replied for the visitors with a try following a penalty line-out.
Tommy Bowe's try, converted by Wallace, put Ulster further in control before Andy Newman went over for Glasgow.
Glasgow had started brightly, being in control for the first 11 minutes during which a lineout drive, a break by Andy Henderson and a Justin Va'a charge-down came to nothing.
There was a setback for Ulster in the sixth minute when influential scrum-half Isaac Boss was forced off with an injury and replaced with Kieran Campbell.
But Ulster took the lead with their first chance, a break by Bowe and Wallace's follow-up leading to Best's try.
Bowe had claims for a try denied when he was pulled back for a forward pass.
And Glasgow, who dominated possession throughout, threatened to come back into the game when a Hefin O'Hare kick led to a penalty in front of the posts, easily converted by Parks.
But Best went over and, although Wallace missed that conversion, the kicker did add to the tally after a sweeping counter-attack led to another score for Bowe.
A forceful thrust led to Glasgow's late try after good work from Rory Lamont and Sam Pinder.
But Colin Gregor conversion attempt drifted wide and, along with it, Glasgow's hopes of a fightback.
Glasgow Warriors: C Gregor, R Lamon, H O'Hare, A Henderson, T Evans, D Parks, G Beveridge, J Va'a, F Thomson, E Murray, D Turner, A Kellock (capt), A Wilson, J Barclay, J Beattie. Replacements: E Milligan, K Tkachuk, A Newman, J Eddie, S Swindall, S Pinder, S Barrow.
Ulster: M Bartholomeusz; T Bowe; A Trimble, K Maggs, P McKenzie; P Wallace, I Boss; B Young, R Best, S Best, capt, J Harrison, M McCullough, N Best, K Dawson, R Wilson.Replacements: P Shields, J Fitzpatrick, T Barker, N McMillan, K Campbell, A Larkin, P Steinmetz.