Fly-half Aj MacGinty added 11 points while winger Niyi Adeolokun scored a crucial try to propel Connacht to their maiden appearance in Pro12 final after defeating the defending champions Glasgow 16-11 in the play-off at Galway.
Pat Lam's men were pushed hard by Glasgow after they came back from 10-3 at the end of the first half with lock Leone Nakarawa scoring the try in the second half.
Connacht now have won the Scottish side on all the three occasions in this season with the previous two victories coming at the group stage.
It was in the 24th minute, MacGinty scored the first point of the game with a successful penalty kick to which Weir responded back with a three-pointer for Glasgow.
With the score levelled at 3-3, Adeolokun try proved to be crucial for Connacht and with conversion from MacGinty, the Irish side had a 10-3 lead over Glasgow.
Glasgow made a strong comeback in the second half with Nakarawa breaching Connacht's defence and scored the first try for his team. Another penalty from Weir in the 56th minute brought down the deficit to just two.
Connacht will be ruing their missed opportunities after they opted for corners in two instances but when Ryan Grant was penalised for driving in on the angle, MacGinty scored the decisive penalty kick which gave them a 16-11 victory over Glasgow.
Connacht will now take on Leinster in the final of the Pro12 tournament, which will be held at BT Murrayfield on 28th May.