Phoenix were quick off the mark with Luke Fulghum scoring the opening goal with just 65 seconds on the clock.
Edinburgh killed a long Phoenix power play after Mark Garside caught Kyle Bruce with a high stick, the Phoenix forward needing stitches to a facial wound.
Phoenix dominated the first period and doubled their lead in the 17th minute with an excellent finish from Nathan Ward. Edinburgh staged a late rally and were unlucky when a shot from Andrej Rajchak hit a post just before the first break.
The second period saw Phoenix extend their lead with Mulherin scoring on his return to the line-up before the Capitals pulled a goal back in bizarre fashion.
Rajchak drove to the net and the attempted clearance by Phoenix net minder Stephen Murphy rebounded past him off team-mate Kenton Smith.
Edinburgh made it a one-goal game in the 45th minute Iain Bowie the scorer as Phoenix began to run into penalty trouble.
Inevitably, though, as the visitors pressed for an equaliser Phoenix got the breakaway and leading scorer David-Alexandre Beauregard duly netted what proved to be the game-winning goal.
Edinburgh tried to stage a grandstand finish as Mark Hurtubise scored a late third for the visitors but Phoenix held on for a valuable two points.
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