Three Glasgow tries, two just before and one just after half-time, had stretched the home side's lead to 24 points with almost all of the second half to go, Glenn Bryce scoring the pick with a fine step inside off his left foot.
However, the Connacht front row started to come into its own, setting up scrum-half John Cooney before winning a penalty try.
A perfect Rory Parata kick through was then gathered by Danie Poolman for the Irish side's third, before Nepia Fox-Matamua rolled off the back of a maul to dot down.
The conversion pulled Connacht to within a point, but they couldn't find a way through, and Glasgow had weathered the storm, but only just.
The Warriors' Week 1 conquerors, Scarlets, went top of the GUINNESS PRO12 table and sent Ulster home pointless, as they secured a 22-12 victory.
It was all Dan Jones penalties before half-time as the visitors failed to get on the scoreboard, but the game sparked into life after the break, Scarlets winger Hadleigh Parkes scoring in the corner.
Ulster responded, capitalising on John Barclay's yellow card to get a try of their own through Rory Scholes, before a penalty try and Scarlets' third yellow card of the game brought the away side to within a score.
However, the 14-man Scarlets held on, an Aled Thomas penalty with the clock red denying Ulster even a losing bonus point.
Edinburgh made it two wins from two, beating Benetton Treviso 27-24 in Italy.
First-half tries for Phil Burleigh, Damien Hoyland and Sam Beard put Edinburgh in the driving seat, although Treviso's Andrea Pratichetti pulled one back to make it 24-16 at half time.
The Italian side scored again straight after the resumption, Duncan Naude forcing his way down in the corner, and a Jayden Hayward penalty pulled the scores level.
However, Greig Tonks kept his cool with eight minutes left to slot the penalty and secure a valuable win.
A determined Cardiff Blues side almost upset the applecart at the RDS, as Leinster eventually turned possession into points to win the game 23-15.
Rhys Patchell's five excellent penalties kept the Blues in a game in which they were second best, but Jack Conan's score and a late penalty try ensured Leinster were deserved victors.
The pressure eventually told, as Leinster, having camped in the corner for five minutes, were awarded a penalty try with four minutes left.
Munster were the last side to beat Ospreys at home and they repeated the feat with a 20-18 victory at the Liberty Stadium.
A slow-moving first 40 minutes saw three Sam Davies penalties give Ospreys a slender lead, but second-half tries for Stephen Fitzgerald and CJ Stander were enough to grab Munster the win.
And while Newport Gwent Dragons couldn't match the 61 points of Cardiff last week, they still got a win against Zebre, scoring all their points in the first half-hour and emerging 13-0 winners.
TRY OF THE WEEKEND
Ospreys' first GUINNESS PRO12 defeat at home in 14 matches was never going to come easily, and Stephen Fitzgerald's connection with Tyler Bleyendaal proved decisive.
12-6 down early in the second half, Bleyendaal, making his first start for Munster, picked out Fitzgerald all alone on the left wing with a precision cross-kick.
In a game with try-scoring opportunities at a premium, it was exactly the spark that Munster needed to take the lead.
BREAK OF THE WEEKEND
There wasn't a lot to celebrate for Zebre again this weekend, but they did produce at least one flowing move, which would have been a try-scoring one had the referee not blown up, with the ball appearing to be unplayable.
At 13-0 down, Dries van Schalkwyk drove at the Dragons' 22, off-loading with his wrong hand to Paul Derbyshire, who in turn passed out of the tackle to hooker Tommaso d'Apice.
Luke Burgess was on his shoulder, but was hauled down inches short of the line, and Zebre remained scoreless, as they would for the rest of the game.
PLAY OF THE WEEKEND
Rory Scholes wouldn't have thanked his scrum-half Paul Marshall for the pass that hit him standing still with a tackler in his face, and still 12 metres from the line.
However, the six foot two winger stepped inside the tackle of Tom Williams, slipped through the arms of Aled Davies, and then powered his way over the line with flanker James Davies hanging from his waist.
At 16-0 it was a lifeline for Ulster, and though it would prove to be in vain, it was nevertheless a remarkable finish from the youngster.
PLAYER OF THE WEEKEND
Rhys Patchell cashed in last week to score three tries in Cardiff's rout of Zebre, but it was his kicking boots that won him plaudits this round.
Patchell's side were getting pummelled all over the pitch by Leinster, but when the home side gave away a penalty anywhere near their own half, it was worth three points to Cardiff.
The 22-year-old fly-half sent penalties over from all over the pitch, with neither distance or accuracy proving an issue.
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