A well-worked try from Andrew Trimble and eight points from the boot of Paddy Jackson proved not quite enough to see Ulster over the line, as the Top 14 side picked them off with three tries in the last 10 minutes.
The visitors had edged it in the first half with good possession and exemplary organisation, but seemed to tire towards the end of a second period where Bordeaux took complete control and threw everything at them.
Trimble, captaining the side from the wing on his first outing of the season, was one of four changes to the backline which went down 30 - 25 in Connacht last weekend. A brand new centre pairing of Luke Marshall and Stuart Olding brought an explosive blend of muscle and guile, while Ruan Pienaar also returned to the starting line-up to direct play from scrum-half. Full-back Jared Payne, wing Craig Gilroy and out-half Jackson all retained their places to complete the line.
Four further changes up front saw Andy Warwick selected at loosehead alongside Rory Best and Rodney Ah You, a new second row of Alan O'Connor and Franco van der Merwe, and Sean Reidy replace Roger Wilson at Number Eight, flanked by Iain Henderson and Clive Ross.
Perhaps the most intriguing aspect of the clash was the battle between the two sides' out-halves, Paddy Jackson and Ian Madigan, prime pretenders to Johnny Sexton's place as playmaker in the Irish national team.
Madigan drew first blood with a third-minute penalty, but cohesive defence from Ulster as the French side tried to force over a rolling maul kept the scoreboard static - although a poor blind pass from Reidy almost let Blair Connor in but for good covering from Jackson and Warwick.
The Ulster score on 20 minutes came almost from nowhere, as Pienaar capitalised on some sluggish defence to pick out Trimble's run right through the heart of the Bordeaux rearguard, slow to react as Ulster recycled exceptionally quickly.
The visitors went on to rule the roost for the remainder of the first half, bossing successive scrums and mauling well in the Bordeaux '22' on the half-hour until the opening presented itself - Trimble again involved, this time on his wing - although his final pass to Jackson was too far from the out-half's grip for him to collect cleanly as he bore down on the line.
Ulster came even closer on 36 minutes, a desperate grab of Pienaar's jersey the only obstacle to the scrum-half breaking free to score, then Warwick fumbling forward as he stooped to ground after more speedy recycling. A Jackson penalty followed after Bordeaux contrived to surrender possession at the resulting scrum, sending Ulster off at the break a fully-deserved seven points ahead.
Half-Time Score Bordeaux-Bègles 3 Ulster 10
The breaking news of Anthony Foley's tragic passing gave the second half a somewhat sombre tone, and an improving Bordeaux crept their way up the scoreboard courtesy of a brace of penalties, from Madigan on 43 minutes and his replacement Lionel Beauxis 10 minutes later.
Ulster did well to weather the storm with no further concessions, and took their chance fantastically on 56 minutes when Jackson slotted over an impressive penalty kick from just short of half-way. Normal service then resumed with Bordeaux applying severe pressure, in particular through two powerful rolling mauls which Ulster held at bay long enough to eventually make critical steals.
The hosts came even closer on 65 minutes, finding Jean-Marcelin Buttin wide on the right after good phase play from the forwards, but the full-back was bundled out of play just in time by a combination of Olding and Gilroy.
They should have taken the lead five minutes later when Jean-Baptiste Dubié inexplicably knocked on inches from the try-line, but had better fortune as huge replacement prop Sébastien Taofifenua just grounded a split second before being dragged into touch by three Ulster tacklers.
Beauxis' conversion missed but it mattered little as a fabulous surge from Metuisela Talebulamaijaina straight from the restart ended with the award of a penalty try, Reidy penalised and sinbinned for illegally preventing the try by engaging in the tackle before the ball recipient - Baptiste Serin in this case - took possession of the ball.
A late breakaway try from Connor rubbed salt into the Ulster wounds and perhaps flattered the hosts slightly, but Ulster must regroup quickly as next Saturday's visit of Exeter Chiefs to the KIngspan is now a must-win encounter.
Full-Time Score Bordeaux-Bègles 28 Ulster 13
Bordeaux-Bègles (15 - 9) Jean-Marcelin Buttin; Adam Ashley-Cooper, Jean-Baptiste Dubié, Romain Lonca, Blair Connor; Ian Madigan, Yann Lesgourgues
(1 - 8) Jefferson Poirot, Clement Maynadier, Vadim Cobilas, Jandré Marais (c), Cyril Cazeaux, Luke Jones, Hugh Chalmers, Loann Goujon
Replacements (16 - 23) Ole Avei, Sébastien Taofifenua, Marc Clerc, Tom Palmer, Marco Tauleigne, Baptiste Serin, Lionel Beauxis, Metuisela Talebulamaijaina
Ulster (15 - 9) Jared Payne; Andrew Trimble (c), Luke Marshall, Stuart Olding, Craig Gilroy; Paddy Jackson, Ruan Pienaar
(1 - 8) Andy Warwick, Rory Best, Rodney Ah You, Alan O'Connor, Franco van der Merwe, Iain Henderson, Clive Ross, Sean Reidy
Replacements (16 - 23) Rob Herring, Kyle McCall, Ross Kane, Pete Browne, Roger Wilson, Paul Marshall, Jacob Stockdale, Rob Lyttle
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