Glasgow resisted a fight back from Connacht to seal their first victory of the Pro12 season, winning 33-32 at Scotstoun Stadium on Friday.
Warriors led 30-6 early in the second half but the visitors the scored four unanswered tries to set up a grandstand finish.
Despite a bright start by the Irish province, as they went into a 3-0 lead via the boot of Jack Carty, the Scottish outfit dominated the rest of the first half.
Rory Clegg kicked three penalties while Glenn Bryce and the impressive Scot Cummings went over for a 23-3 advantage.
Carty's three-pointer reduced the deficit but Rob Harley's touchdown in the second period gave the hosts a comfortable buffer.
That seemed to seal the win but Connacht came back into the contest with scores from John Cooney, Danie Poolman and a penalty try.
Clegg's late penalty eased the tension for the hosts, although Nepia Fox-Matamua's touchdown gave the away team two deserved bonus-points.
Following Glasgow's opening round 16-10 home defeat to the Scarlets in the Pro12, the defending champions needed to respond.
However, they were on the back foot in the early exchanges and Connacht scored first when Carty landed a penalty.
That deficit didn't last long as Gregor Townsend's men hit their stride and successive efforts off the tee from Clegg took the Warriors 6-3 ahead.
The fly-half then added another three-pointer before a period of pressure close to the opposition line resulted in Bryce scampering over.
Glasgow were now well on top and a try for Cummings, after further good work from the forwards, characterised their dominance.
A second penalty from Carty – the final act of the half – just about kept the visitors in the contest, but Harley's score minutes into the second period appeared to have extinguished any thoughts of a comeback.
To the Irish province's credit, they continued to fight and Cooney finished off a well worked set-piece move.
Glasgow prop Zander Fagerson then received a yellow card and that was to prove crucial as referee Ian Davies subsequently awarded a penalty try.
Pat Lam's side were soon within five points when Poolman latched onto an intelligent grubber through, but they couldn't quite find the decisive score.
Instead, the Warriors lifted the intensity once more and Clegg's penalty secured a narrow win.
There was some late panic for the hosts through Fox-Matamua's try but Glasgow managed to hold on.
For Glasgow Warriors:
Tries: G Bryce, Cummings, Harley
Cons: Clegg 3
Pens: Clegg 4
Yellow card: Fagerson
Tries: JCM Cooney, Penalty try, Poolman, Fox-Matamua
Cons: Carty 3
Pens: Carty 2
Glasgow Warriors: 15 Glenn Bryce, 14 Junior Bulumakau, 13 Peter Murchie (c), 12 Fraser Lyle, 11 Lee Jones, 10 Rory Clegg, 9 Mike Blair, 8 Adam Ashe, 7 Chris Fusaro, 6 Rob Harley, 5 Kieran Low, 4 Scott Cummings, 3 Mike Cusack, 2 Pat MacArthur, 1 Alex Allan
Replacements: 16 Kevin Bryce, 17 Jerry Yanuyanutawa, 18 Zander Fagerson, 19 James Eddie, 20 Tyrone Holmes, 21 Hugh Blake, 22 Grayson Hart, 23 Scott Wight
Connacht: 15 Tiernan O'Halloran, 14 Danie Poolman, 13 Rory Parata, 12 Bundee Aki, 11 Matt Healy, 10 Jack Carty, 9 John Cooney, 8 Eoghan Masterson, 7 Nepia Fox-Matamua, 6 John Muldoon (c), 5 Andrew Browne, 4 Quinn Roux, 3 Rodney Ah You, 2 Tom McCartney, 1 Denis Buckley
Replacements: 16 Dave Heffernan, 17 JP Cooney, 18 Finlay Bealham, 19 Ultan Dillane, 20 Ben Marshall, 21 Kieran Marmion, 22 Craig Ronaldson, 23 Fionn Carr
Venue: Scotstoun Stadium, Glasgow
Referee: Ian Davies (Wales)
Assistant Referees: Sam Groves-White (Scotland), Mike Adamson (Scotland)
TMO: Iain Ramage (Scotland)