Evans tore through a gap in the Connacht defence, causing the initial damage and was then brilliantly fed by out-half Dan Parks for a short run-in to the line, with just two minutes gone.
Parks nailed the conversion which Connacht out-half Ian Keatley was able to cancel out with a well-struck drop goal after Frank Murphy and Mike McCarthy had set the visitors up for a series of attacks. Either side of that score, Parks missed two penalty attempts.
Towards the end of the first quarter, Parks and Keatley swapped penalties but Glasgow continued to look the more threatening with ball in hand and particularly in the backs, with Argentinian full-back Bernardo Stortoni almost released for a try.
The second Glasgow touchdown duly arrived on 26 minutes when number 8 Johnnie Beattie picked off the base of a solid scrum and powered his way over from close range, with Parks adding the extras for a 17-6 advantage.
A third converted try, this time from Thom Evans, put 18 points between the sides. Experienced scrum half Chris Cusiter combined with Parks whose popped pass set Evans free and the winger finished with aplomb, fending off Gavin Duffy to score from 40 metres out.
Connacht's defence, up front and out wide, was struggling to cope as the Evans revelled in the space offered to them and the likes of returning lock Richie Gray continually made the hard yards.
Credit to Michael Bradley's men though, they kept making the tackles and two quick-fire penalties from Keatley, just before half-time, gave them some heart at 24-12 down.
Keatley and hooker Sean Cronin were the only changes to the Connacht side that secured a famous Challenge Cup win in Montpellier last weekend.
Former England prop Robbie Morris, back from suspension, was introduced by Bradley for the second half and Keatley raised the flags once again to reduce the deficit to nine points.
He missed with his next attempt, after Parks had nudged Glasgow onto 27 points, but despite losing replacement Andrew Browne to the sin-bin, Connacht rallied and hit back with a try from top-scoring winger Fionn Carr.
Carr has been scoring tries for fun of late, this was the Ireland 'A' international's eighth try in seven matches in all competitions this season.
As Glasgow broke forward, a pass from Stortoni was intercepted by scrum half Murphy who quickly set the pacy Carr up for a run to the far corner, avoiding Graeme Morrison's despairing tackle along the way. Keatley missed the conversion.
The second half was a scrappy affair and Glasgow let the tempo drop as Connacht's committed defence, typified by hard-hitting skipper John Muldoon, kept them at bay.
However, any thoughts of a comeback win for the visitors were firmly put to bed when Thom Evans pounced on a Mark McMillan chip to collect the Warriors' bonus point try, with six minutes remaining.
Replacement Colin Gregor added the conversion for good measure, capping off a result which the home players merited for their first half efforts.
Sean Lineen's charges should have had the game wrapped up by the hour mark, but Connacht dug their heels in, turned it into a dogfight and will be disappointed that they failed to emerge with a losing bonus point.
|Glasgow Warriors Score Card|
|Connacht Rugby Score Card|