The Ospreys beat Connacht 24-20 in Galway on Saturday but missed out on a bonus point, meaning they will travel to Munster in the Pro12 semi-finals.
The Welsh region produced an outstanding first half performance and moved to within touching distance of a home semi-final after going into the break 24-0 in front.
They scored three converted tries through Dan Biggar, Ben John and Rhys Webb as they effectively sealed the win by half-time.
All the Ospreys needed after the interval was another touchdown but they failed to gain that crucial score as the Irish province controlled the second period and reduced the deficit through Eoghan Masterson and Dennis Buckley tries.
Steve Tandy's men went into the game at the top of the Pro12 table but knew any slip-up could cost them the chance of a game at the Liberty Stadium in the knockout stages.
A victory was therefore crucial to their title ambitions and they got the ideal start through Biggar's accuracy from the tee.
Although Pat Lam's side briefly threatened in the visiting 22, the Ospreys were very much in control and a superb run by Ben John resulted in their first try.
Connacht's defence was already under pressure but John took the ball and weaved his way through some weak tackling before he off-loaded to his stand-off to touch down.
The outside centre then went from provider to scorer as he took a neat line to go over unopposed for a 17-0 lead.
With the momentum with the away team, they crossed for the third time as another superb move resulted in Webb scampering over.
It was a poor first-half performance by the hosts but, to their credit, they lifted their display significantly after the break.
Two Jack Carty three-pointers got the scoreboard rolling before Masterson went over.
Ospreys tried to gain a foothold but Connacht remained resilient and deservedly went over for a second time through Buckley in the last minute.
Tries: Masterson, Buckley
Cons: Carty 2
Pens: Carty 2
Tries: Biggar, John, Webb
Cons: Biggar 3
Connacht: 15 Tiernan O'Halloran, 14 Fionn Carr, 13 Robbie Henshaw, 12 Bundee Aki, 11 Matt Healy, 10 Jack Carty, 9 Kieran Marmion, 8 Eoin McKeon, 7 Eoghan Masterson, 6 John Muldoon (c), 5 Aly Muldowney, 4 Andrew Browne, 3 Rodney Ah You, 2 Tom McCartney, 1 Denis Buckley.
Replacements: 16 Dave Heffernan, 17 JP Cooney, 18 Finlay Bealham, 19 Michael Swift, 20 George Naoupu, 21 John Cooney, 22 Miah Nikora, 23 Shane O'Leary.
Ospreys: 15 Dan Evans, 14 Hanno Dirksen, 13 Ben John, 12 Josh Matavesi, 11 Eli Walker, 10 Dan Biggar, 9 Rhys Webb, 8 Dan Baker, 7 Justin Tipuric, 6 James King, 5 Alun Wyn Jones (c), 4 Tyler Ardron, 3 Dmitri Arhip, 2 Scott Baldwin, 1 Nicky Smith.
Replacements: 16 Sam Parry, 17 Marc Thomas, 18 Aaron Jarvis, 19 Rory Thornton, 20 Dan Lydiate, 21 Tom Habberfield, 22 Sam Davies, 23 Jonathan Spratt.
Referee: Marius Mitrea (FIR)
Assistant Referees: Andy Brace, Nigel Correll (IRFU)
TMO: Seamus Flannery (IRFU)