The 1999 European Champions made amends for a Magners League defeat to the same opposition at Ravenhill earlier this season, as well as a Heineken Cup reverse at Murrayfield back in October.
Tries from centre Darren Cave and man-of-the-match Isaac Boss gave the hosts a 15-6 advantage 55 minutes into a hard-fought Pool 4 encounter after Edinburgh had led 6-5 after a first-half in which they had a considerable wind behind them.
Edinburgh hit back through a Jim Thompson try with just over an hour played but Ian Humphreys' second penalty of the match and another from replacement Nial O'Connor secured victory and prevented Edinburgh from gaining a losing bonus-point.
Ulster are now level on points with French giants Stade Francais at the top of the pool, although the Parisians can secure top spot with a game to spare if they record a bonus-point win at home to Bath on Saturday afternoon.
If Stade fail to do so, then Ulster could still win the pool if they can defeat Bath at The Rec and Edinburgh do them a favour against Stade in the Scottish capital.
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