Connacht trailed by just one point, 20-19 at the break, thanks to a superb late try from Ted Robinson which Eric Elwood converted and the young winger nearly stole it at the death when his searing touchline run was stopped just metres from the line.
On balance Glasgow probably deserved their victory but credit Connacht for the way in which they refused to buckle. An early second half try from Gareth Flockhart was converted by Tommy Hayes to stretch Glasgow's lead to eight points but eric Elwood landed a penalty to keep his side very much in the hunt. Another Glasgow try seemed certain to kill off the home challenge but back they came to score a superb try from Wayne Munn and Elwood's conversion left just five points separating the sides.
Connacht continued to press but the Scots held on and will now meet Leinster in the semi final in Lansdowne Road next weekend..