Hefin O'Hare touched down for Glasgow, who trailed by 18-8 at half-time with Ruaridh Jackson in the sin-bin.
Connacht crucially added two more tries early in the second half while Jackson was off, and the icing on the cake was a 62nd-minute penalty try
Eric Elwood's charges set a brisk early tempo. A good spell of ball retention drew an offside decision against Glasgow and the impressive Ian Keatley converted the resulting kick.
The Connacht out-half missed a more difficult penalty, three minutes later, but the hosts continued to press with Mike McCarthy and Michael Swift imposing themselves up front.
In the ninth minute, Connacht showed patience on the attack before Keatley went on the outside and passed for Tuohy to use his strength to crash over.
8-0 down, Glasgow hit back from their very first attack. Their captain Colin Gregor took a quick tap from a midfield penalty, noticing an overlap on the right, and Bernardo Stortoni produced the final pass for O'Hare to finish off.
The try was unconverted and Connacht wasted little time in responding. Keatley used his weaker left boot to grubber through for full-back Duffy to race up and score in the corner.
Keatley's successful conversion was cancelled out by Jackson's only penalty of the night, midway through the half.
Glasgow threatened the Connacht line in the lead-up to the break. A knock on from Gregor saw the Warriors stumble within scoring range, and Robert Harley was also hauled down a few metres out.
But Keatley kicked Connacht ten points clear after Jackson was yellow carded for tackling Tuohy without the ball as the winger chased his own kick.
The second half burst into life with two tries in a four-minute spell. Keatley spotted some space and put Fionn Carr away for a well-taken third try.
Now brimming with confidence, Connacht claimed a cracking bonus point score. Prop Brett Wilkinson charged forward on a 20-metre break, passed for Keatley and the out-half's looping delivery sent Tuohy thundering over for his second try.
Keatley converted one of the tries, leaving Glasgow 30-8 adrift. The Scots were camped in the Connacht half for much of the remainder but just could not find a way past the hosts' defence, which was marshalled superbly by skipper Johnny O'Connor.
Connacht's dominance of the set piece was confirmed by the penalty try, with replacement Rodney Ah You performing well in the front row.
With their first ever season's double over Glasgow in the bag, Connacht saw out the final minutes in their own half and down to 14 men following the sin-binning for Shane Monahan.
|Connacht Rugby Score Card|
|Glasgow Warriors Score Card|