Only a point separated the teams going into the final ten minutes, but Ulster got their hands on the ball, worked their way up to the line before Coetzee powered over.
Billy Burns converted, and there was no way back for Connacht, with Ulster now set to travel to Glasgow to take on the Warriors in the semi-final.
On an emotional day in Belfast, with Rory Best playing his final game at the Kingspan Stadium, Nick Timoney had given Ulster the early lead.
Bundee Aki responded for Connacht, aiming for a second win in a row in Belfast, but in the end they left themselves too much to do.
Ulster were dominant in the opening stages, and John Cooney put them in front with a simple penalty after Colby Faingaa was pinged at the breakdown.
On 15 minutes they increased their lead with a stunning try. Having punched away through the forwards, Coetzee stepped out of one tackle, offloaded to Kieran Treadwell, who did the same and Timoney finished like a winger, stepping inside Matt Healy on his way to the line. Cooney’s conversion was off-target and Ulster led 8-0.
Connacht had struggled to land a blow to that point, but a beautiful wide pass from Jack Carty tempted Robert Balocoune but drifted outside him to put Healy away. The Ulster scramble defence got back to stop the try, but Treadwell was slow to roll away and Carty slotted the resulting penalty.
Still, it was Ulster who finished the half on the front foot, working their way up to the Connacht line. The visiting defence held and forced Dan McFarland’s team right back out of the 22. However Aki was penalised for a dangerous tackle on Burns, and Cooney slotted the three to make it 11-3 at the break.
Another interception proved even more costly for Ulster. Faingaa raced onto a pass from Burns, slipped a tackle and raced clear. He then found Aki with a clever one-handed offload and he had the pace to outstrip the cover and dive under the posts. Carty converted from in front.
The teams then exchanged penalties to leave the score at 14-13 with ten minutes to go. Best having taken his leave after a typically all-action display.
That set up a grandstand finale, and it was Coetzee, the man of the match on the day, who delivered the final blow to set up a date with Glasgow Warriors at Scotstoun.
The result brings the curtain down on a successful season for Connacht which has set up a return to Champions Cup rugby for the province. But unfortunately a semi final appearance proved one step too far for Andy Friend’s side this season.