Tries either side of halftime from wingers Mike Allen and Craig Gilroy proved decisive as the Warriors were outpowered in Gregor Townsend's first game in charge.
They were granted a lifeline through Tommy Seymour's try against his former club which brought them to within five points with 11 minutes remaining.
But Niall O'Connor, on his second debut for the club after returning from Connacht, made the game safe with a late penalty although he missed 11 points with the boot.
The most impressive of Ulster's four debutants was the bruising Kiwi No.8 Nick Williams who made constant ground from the backrow.
With Ireland internationals Tommy Bowe and Stephen Ferris still to return, having been given extra time to recover from their summer in New Zealand, the win will give the New Zealander Anscombe plenty of confidence heading to the Liberty Stadium to face the champion Ospreys.
However he may have to make do without Ireland U20s captain Paddy Jackson - Ulster's surprise starter in last year's Heineken Cup final - who was injured early, making way for O'Connor, who slotted a penalty from the Glasgow 10m with his first touch to open the scoring in the 12th minute.
Following a good spell in possession Glasgow answered on 21 minutes with a penalty - kicked from short range by their own rising star at fly-half, Duncan Weir.
Ulster responded immediately - a booming kick from O'Connor put his side within striking distance before a series of thunderous drives by back-row powerhouses Robbie Diack and Williams put the No.8 over the line only for the ball to be dislodged by a Tom Ryder tackle.
The Irish side kept Glasgow pinned deep within their own half - thanks in part to huge hits by Diack - and eventually the pressure led to another penalty.
O'Connor, however, was unable to convert, hitting the post from 30 metres before missing entirely with on a straightforward attempt moments later.
Talismanic captain Johann Muller forced another penalty at the breakdown, but turned down a shot at goal from 40m in favour of going to the corner with the ensuing rolling maul driving within 10 metres before Glasgow skipper Al Kellock infringed and was sent to the sin bin.
With two minutes until half-time, Ulster got their breakthrough, Allen scoring in just his second competitive start after fit-again Jared Payne switched the play back to the short-side and hit the 22-year-old wing with a perfectly weighted cut-out pass.
Duncan Weir saw his speculative drop-goal attempt drift wide in the last action of the half but his side went in still in touch with the score at 8-3
The introduction of Scotland scrum-half Chris Cusiter at half time sparked a strong start by Glasgow - until Ulster's Gilroy broke off his wing and picked off an interception to run in untouched from 50 metres - with O'Connor's conversion stretching the lead to 12 points.
As Townsend called on a bench boasting six internationals, Ruaridh Jackson, on at fly-half hit the post with a penalty attempt before putting through ex-Ulsterman Seymour for a quality try, and adding the touchline conversion to make it 15-10 with ten minutes to play.
O'Connor's tactical kicking kept the Scottish side out of range in the closing stages, and the Ulster replacement fly-half sealed the result with a last-minute penalty.
|Ulster Rugby Score Card|
|Glasgow Warriors Score Card|