Connacht's recent winning streak was ended by Glasgow Warriors after a narrow loss at Scotstoun.
Andy Friend's side, who travelled to Glasgow after winning six on the bounce in the BKT United Rugby Championship, will travel to Ulster in the last eight on the weekend of 5/6 May.
They had expected to need a win to qualify for the Champions Cup but Munster's draw with the Sharks earlier on Saturday means the Westerners will qualify unless the South Africans win the United Rugby Championship or Scarlets or Benetton win the Challenge Cup.
Glasgow, who will host Munster in the quarter-finals, were already assured of fourth place in the URC and started the match like they did not have a care in the world.
They fired through several slick phases straight from kick-off, with Huw Jones and Stafford McDowall making good headway with powerful midfield bursts.
Then Rory Darge appeared on the scene to send Jones over for the game's opening try with neat back-handed offload.
Connacht bounced back immediately by forcing Jones to carry the ball back over his own line, and then pressurising the home scrum into conceding a penalty try.
Warriors’ scrum continued to struggle, but their play around the park was razor sharp, and they raced back into the lead in the 18th minute when Tom Jordan juggled the ball before streaking under the posts.
The home side lost Scott Cummings to the sin bin for an obstruction on Caolin Blade and Jack Carty kicked the three easy points for the visitors.
Connacht worked hard to take advantage of the extra man and were rewarded with the lead when No 8 Cian Prendergast burrowed over from a tap penalty, with Carty adding the extras.
Warriors did not panic however, and they struck back before Cummings had returned to the fray with Ali Price finishing off on the left after some sharp link play.
Cole Forbes thought he had scored try number three for Glasgow – who led 19-17 at the break – after Jordan and Jones capitalised on turnover ball, but the latter had knocked-on while being tackled and so it was chalked off.
Connacht were forced into a triple change at half-time, with Denis Buckley, Finlay Bealham and Prendergast all replaced.
There was no more scoring until the hour mark, when replacement Glasgow hooker Johnny Matthews went over at the back of a lineout maul and George Horne added the touchline conversion to make 26-17.
Connacht would not lie down though, and after Horne and Carty had exchanged penalties, a Shamus Hurley-Langton try converted by Carty made for a tense final five minutes – but Warriors held on.
BKT United Rugby Championship quarter-finals
- Leinster v Cell C Sharks
- Ulster v Connacht
- DHL Stormers v Vodacom Bulls
- Glasgow Warriors v Munster
Games to be played on weekend of 5/6 May
