Scrum half Murray, who helped Ireland U-20s beat Italy last weekend, collected a smashing try which broke UCD hearts.
Two penalty goals from the ever-reliable Conor Kilroy had Garryowen leading 6-0 at half-time.
Kilroy punished another UCD error with his third penalty success before former Ireland underage international Noel Reid split the Garryowen defence to score a fine try that he converted himself before Murray grabbed the crucial try to give his side the points.
Winners at Old Belvedere last time out, St Mary’s College made the most of their scoring opportunities to close back in on one of the four play-off positions.
However, Blackrock will feel this is one that got away. The division’s top try scorer, Shane Monahan, missed two gilt-edged try-scoring chances when he knocked on twice with the line in his sights.
Shaun McCarthy and Darragh Fitzpatrick exchanged penalties before St Mary’s winger Darragh Fanning struck for an unconverted try before half time.
McCarthy kicked Mary’s 11-3 ahead and when lock Conor McInerney scored a crucial 68th minute try, the game was as good as over.
’Rock had little in reserve but they did gain a consolation try in injury-time when Monahan skipped clear of the cover to score a five-pointer.
The Limerick derby went as expected when Shannon proved too strong for lowly Young Munster, who remain rooted to second last place in the table.
Old Belvedere dealt UL Bohemians a severe blow when they escaped with the spoils from a tight game at Annacotty, and at Stevenson Park, Ballymena shaded the Ulster derby by beating Dungannon 19-15.
The challenge for promotion from division two remains intense, with leaders Lansdowne, second placed UCC and third placed Ballynahinch all winning.
Lansdowne had to fight to get the better of Thomond at Fitzgerald Park before emerging with a 9-0 win, UCC found the going tough against Bruff before winning 14-6 while Ballynahinch had a facile 46-15 win over Bective Rangers at Donnybrook.
In Division Three, Midleton continued a winning run when they proved too strong for Suttonians at Towns Park, the home side winning 17-12, but they got no favours from the three teams above them. Corinthians, County Carlow and Queens University all emerged with deserved victories from their respective games against Banbridge, Portadown and Barnhall.
At the bottom, it was no surprise that Sunday’s Well won against Waterpark at Ballinakill, but the Cork side had to fight a lot harder than anticipated before securing a 15-10 win. That, combined with Banbridge losing to Corinthians, gives Sundays Well a little breathing space, having jumped from 15th to 14th.