Gunners coach Michael Bradley opted for a reshuffle, with stand-off Phil Godman and No. 8 Stuart McInally the only players to retain their places following the 17-12 defeat by Glasgow Warriors on New Year's Day.
Ulster, who welcomed back Ireland quartet Trimble, Best, Stephen Ferris and Tom Court took an early lead through Ian Humphreys' fourth minute penalty, and eight minutes later he kicked the visitors six points ahead before being taken off with a serious-looking injury.
Edinburgh responded though through Scotland international Nick De Luca, who pounced on a loose ball to go over on 13 minutes with Godman adding the extras.
A Greig Laidlaw penalty four minutes later extended Edinburgh's advantage but that was as good as it got as Ulster turned the screw.
South African scrum-half Ruan Pienaar, who took over kicking duties from the unfortunate Humphreys, brought the Ulstermen within a point with a penalty of his own on 20 minutes.
Eight minutes later Best gave the visitors a deserved lead with a well taken try and then, almost immediately, Dan Tuohy crossed the whitewash. Pienaar was wasteful with the boot on both occasions to leave the score 19-10.
Just as Ulster appeared to be heading into the break with a comfortable lead, a needless penalty allowed Laidlaw to reduce the Gunners' arrears to six points.
The Ulstermen, though, dominated proceedings after the interval, and with Pienaar finding his kicking boots, they quickly amassed an unassailable lead.
His penalty on 46 minutes was quickly followed by Trimble's try, and when the South African added his conversion and another penalty, the score was 32-13 and out of Edinburgh's reach.
Alun Walker barged his way over to give the home side hope but Cave's late try, after great work on the counter attack by Paul Marshall, gave Ulster the decisive bonus-point before Pienaar added another three points with a late penalty.
|Edinburgh Rugby Score Card|
|Nick de Luca||1||5|
|Ulster Rugby Score Card|