Connacht were chasing their best ever finish, but Glasgow Warriors' need was greater as they clinched their spot in the last four of the RaboDirect Pro12 with a convincing victory.
Tries from DTH van der Merwe and John Barclay set the Warriors up to face Leinster in the semi-final at the RDS next Saturday.
Duncan Weir got the home side off the mark after a collapsed scrum 10 minutes in, with a penalty from the right. Connacht proved they were in no humour to roll over for the hosts, though, and they drew level through Miah Nikora after 14 minutes.
The kick came in the middle of Connacht's best spell, but they were unable to add to their total as the home side reclaimed the initiative in the 22nd minute, Weir landing a penalty from halfway.
Four minutes later, Van der Merwe got on the end of some quick passing across the line to dive over in the corner. Credit must go to second-row Richie Gray for his rapid-fire pass, but the ease in which the Canadian flyer crossed will have annoyed the Connacht management.
Weir fluffed the conversion from the touchline, but Warriors held solid through the closing minutes of the half and took an 11-3 advantage in at half-time.
Weir stretched the home side's advantage eight minutes into the second period and Warriors made sure of their semi-final berth when Barclay finished off a push-over try.
Scorers -- Warriors: DTH van der Merwe, J Barclay try each, D Weir 4 pens, R Jackson con. Connacht: M Nikora pen.
Glasgow Warriors: S Hogg; F Aramburu, A Dunbar, G Morrison, DTH van der Merwe (C Shaw 60); D Weir (R Jackson 60), H Pyrgos (C Cusiter 64); J Welsh (R Grant 51), D Hall (P MacArthur 51), M Low (M Cusack 60); R Gray (A Kellock 60), T Ryder; R Harley, C Fusaro (C Forrester 69), J Barclay (yc 76-80).
Connacht: G Duffy; T O'Halloran (J Loxton 56), E Griffin, H Fa'afili, K Tonetti; M Nikora (M Jarvis 56), F Murphy (P O'Donohoe 63, yc 68-77); D Buckley (R Ah You 51), A Flavin (D Murphy 75), R Loughney (S Maguire 65); M Swift (M Kearney 65), M McCarthy; J Muldoon, J O'Connor (R Ofisa 54), G Naoupu.
Referee: P Fitzgibbon (IRFU).
Sunday Indo Sport