After roaring into a three-try lead within a half hour, Glasgow were made to wait 45 minutes for the bonus point try against Connacht on Friday.
It was a dream return for centre Max Evans, making his first start of the season.
In only the the second minute, from a scrum set up by his brother Thom's chip ahead, Evans was on the end of a long miss pass from Dan Parks to go over for Parks to convert.
Connacht hit back and after a couple of dents by Frank Murphy and flanker Mike McCarthy, number 10 Ian Keatley dropped a simple goal only for Parks to restore the margin five minutes later with a 30-metre penalty. Neither side were dominating although Glasgow had the best of the territory but a chip into Glasgow's half resulted in Keatley slotting a 42m shot.
Glasgow upped the pressure, sent Connacht skidding backwards in the scrum and opted to go again on winning the penalty, this time number eight John Beattie bursting off the back for the try.
Parks converted, then after big carries by Richie Gray and Moray Low Thom Evans scorched through from 40 metres out, burst Gavin Duffy's tackle and evened the score with his brother. Parks converted and Glasgow approached half-time 18 points up but a couple of controversial Keatley penalties, the second after Max Evans had almost claimed his second at the other end, kept Connacht in the game at the break.
Into the second half, an exchange of penalties between Parks and Keatley did little to enliven a scrappy second half as Glasgow's first-half sharpness deserted them.
Feisty defence with John Muldoon and Frank Murphy putting in big hits kept Glasgow at bay then a Murphy interception put Fionn Carr away for a try seven minutes from time.
Keatley missed the kick but now Connacht were in sight of salvaging something from the game but it was Glasgow who claimed the bonus as Thom Evans notched his second try when he got on the end of a Mark McMillan chip into the danger zone.
Connacht came close in the dying minutes but had run out of shots to trouble the now confident Glasgow defence.
Tries: M.Evans, Beattie, T.Evans 2
Cons: Parks 3, Gregor
Pens: Parks 2
Pens: Keatley 4
Drop goal: Keatley
Glasgow: 15 Bernardo Stortoni, 14 Colin Shaw, 13 Max Evans, 12 Graeme Morrison, 11 Thom Evans, 10 Dan Parks, 9 Chris Cusiter, 8 Johnnie Beattie, 7 John Barclay, 6 Kelly Brown, 5 Al Kellock (c), 4 Richie Gray, 3 Moray Low, 2 Dougie Hall, 1 Jon Welsh.
Replacements: 16 Pat MacArthur, 17 Kevin Tkachuk, 18 Dan Turner, 19 Richie Vernon, 20 Mark McMillan, 21 Colin Gregor, 22 Hefin O'Hare.
Connacht: 15 Gavin Duffy, 14 Troy Nathan, 13 Niva Ta'auso, 12 Keith Matthews, 11 Fionn Carr, 10 Ian Keatley, 9 Frank Murphy, 8 John Muldoon (c), 7 Johnny O'Connor, 6 Michael McCarthy, 5 Bernie Upton, 4 Michael Swift, 3 Jamie Hagan, 2 Sean Cronin, 1 Brett Wilkinson.
Replacements: 16 Adrian Flavin, 17 Robbie Morris, 18 Andrew Browne, 19 Ray Ofisa, 20 Conor O'Loughlin, 21 Miah Nikora, 22 Tiernan O'Halloran.
Referee: Tim Hayes (Wales)
Assistant referees: David Changleng (Scotland), Bob Nevins (Scotland)
Assessor: Dougie Kerr (Scotland)