Graeme Morrison was a try-scorer for Glasgow
Glasgow (15) 39
Tries: Leonelli, Parks, Kellock, Barclay, Morrison
Con: Parks (4)
Pens: Parks (2)
Connacht (16) 34
Tries: Riordan, Ofisa, O'Connor, Yapp
Pen: McHugh (2), Warwick
Glasgow survived two sin-binnings and a late surge from Connacht to secure a precious bonus-point triumph.
Francisco Leonelli, Dan Parks, Ally Kellock, John Barclay and Graeme Morrison scored tries, while Parks kicked the rest of Glasgow's points.
Dan Riordan, Ray Ofisa and Brendan O'Connor and Darren Yapp all crossed the line for the visitors.
Andy Henderson and Scott Barrow were shown yellow cards as Connacht captured two consolation points.
The visitors started well and drew first blood with an opportunist drop-goal by Mark McHugh.
Connacht doubled their tally with a straightforward McHugh penalty from 30 metres but Parks replied with the boot immediately.
Fit-again Argentinian Francisco Leonelli then burst his way into the match by grabbing the opening try - showing great speed and determination to reach his own hack forward.
Connacht bounced back with Riordan latching on to a perfectly-weighted pass from Yapp. McHugh slotted the conversion.
McHugh followed up with his second penalty but Parks hurtled in for a close-range try and added the extras to leave his side adrift by a single point at half-time.
Within seconds of the restart, Glasgow powered into the lead when skipper Kellock found an extra surge of pace to reach the line at the end of a 20-metre rumble.
It took Glasgow just two more minutes to make sure of the bonus point, Barclay providing the finishing touch to a slick move. Parks added both tricky conversions for good measure.
Centre Graeme Morrison added a superb solo score, leaving three defenders in his slipstream. Parks again chipped the goal.
Ofisa and O'Connor grabbed tries, the second of which came after Warriors centre Henderson was the victim of a strange yellow card verdict.
Barrow was then sin-binned for obstruction and Paul Warwick kicked the penalty.
Parks' penalty steadied the Glasgow ship - only for Yapp to snap up the touchdown which captured two consolation points for the Irish side.
Glasgow : Leonelli, Lamont, Morrison, Henderson, O'Hare, Parks, Beveridge, Tkachuk, Thomson, Va'a, Newman, Kellock (capt), Wilson, Petrie. Barclay.
Replacements: Prescott, Lawson, Turner, Swindall, Pinder, Jackson, Barrow.
Connacht: Riordan; Mostyn, Warwick, Williams, Yapp; McHugh, Keane; Hogan, Fogarty (capt), Sturgess; Gannon, Farley; Muldoon, Ofisa, Rigney.
Replacements: Flavin, Wilkinson, Swift, O'Connor, O'Loughlin, Matthews, Hearty.