Edinburgh are level with Glasgow on 50 points in joint seventh place but neither side can be overhauled by the teams below them with one full round of fixtures remaining.
The two Scottish sides go through to the quarter-finals but both will be away from home in the last eight
On a raucous evening in the capital, Ulster pocketed a 16-10 victory but Mark Bennett’s late try meant all was not lost for the home side.
Edinburgh were playing in front of a capacity crowd for the first time at their new home but they were undone by an Ulster side who defended heroically and their own profligacy.
It was bedlam at times. A frantic opening six minutes saw four players go off injured - two from each side - and a yellow card for Ulster’s Nick Timoney. Amid the bedlam Edinburgh conspired to squander two golden try opportunities.
The first was inside 20 seconds as James Lang went on a searing break from his own 22. Unfortunately for the home side, his pass went astray which was to become a recurring theme in the first half for Edinburgh. Lang was injured in the tackle and hobbled off after two minutes to be replaced by Chris Dean.
Edinburgh lost Stuart McInally to an HIA a couple of minutes later when the hooker felt the full force of Timoney’s shoulder. The high tackle saw the Ulster flanker sent to the sin-bin and the visitors also lost their captain, with Iain Henderson trudging off for an HIA. Ulster centre Stuart McCloskey was invalided out shortly after.
It was harum-scarum stuff but Edinburgh were on the front foot and should have pressed home their advantage when Luke Crosbie made excellent ground only to delay the pass with Damien Hoyland outside him. Crosbie was smothered and the chance was gone.
Ulster had been let off the hook twice and they made the hosts pay for it. Their pack may have been a man down but they managed to force Edinburgh’s full complement back towards their own line and the pressure yielded a try in the corner for winger Rob Baloucoune after substitute centre Stewart Moore’s lovely quick hands.
Nathan Doak converted and Ulster had a lead they barely merited.
McInally returned to the fray after passing his HIA but was forced to hobble off less than a minute later, leading to Dave Cherry replacing him on a permanent basis this time.
Ulster weren’t doing a huge amount in an attacking sense but they had doused Edinburgh’s early fire and Doak dispatched a couple of penalties from around the 35-metre mark to push the visitors 13-0 ahead.
A frustrating half for the home side ended on a more upbeat note when Emiliano Boffelli slotted a penalty just before the break after Ulster had gone offside.
The respite was only temporary. Ulster’s gameplan was simple and direct but Edinburgh were struggling to gain a foothold and were giving away a string of penalties, prompting some heckling of referee Ben Whitehouse by the home fans.
Doak struck the post with one attempt but had more success five minutes later after Pierre Schoeman had been guilty of a shove off the ball.
Edinburgh needed a way back in but Ulster were defending as if their lives depended on it. Some of it bordered on the illegal and Alan O’Connor was sin-binned after one transgression too many as Edinburgh pounded the Ulster line, hoping brute force would open the door.
When the breakthrough finally came it was via a piece of skill rather than a blunt instrument. Bennett spotted an opening and chipped through, collected and touched down in one swift movement to spark uproar in the stadium. Boffelli’s successful conversion had the home fans stamping their feet in appreciation and willing their team on in the final ten minutes.
Scorers: Edinburgh: Try: Bennett. Con: Boffelli. Pen: Boffelli.
Ulster: Try: Baloucoune. Con: Doak. Pens: Doak 3.
Edinburgh: J van der Walt; D Hoyland, M Bennett, J Lang (C Dean 2), E Boffelli; B Kinghorn, B Vellacott (H Pyrgos 58); P Schoeman, S McInally (D Cherry 4-15, 15), WP Nel (L-R Atalifo 61), J Hodgson (P Phillips 62), G Gilchrist, L Crosbie (B Muncaster 49), H Watson, M Bradbury.
Replacements: H Courtney, M Currie.
Ulster: M Lowry (R Lyttle 58); R Baloucoune, J Hume, S McCloskey (S Moore 6), E McIlroy; B Burns, N Doak (J Cooney 50); A Warwick (E O’Sullivan 49), R Herring (B Roberts 66), G Milasinovich (R Kane 61), K Treadwell, I Henderson (A O’Connor 4), M Rea (M Rea 77), N Timoney, D Vermeulen.
Referee: Ben Whitehouse (WRU)