Goals from Chris Turner and Mark Gilhaney put the Sons well in control at the Bet Butler Stadium.
Accies striker Stevie May did reduce the deficit on the hour mark, but a Jonathan Page own goal ensured Murray's side made the most of defeats for relegation rivals Cowdenbeath and Airdrie to finally climb off bottom spot.
Boss Murray said: "It was a great victory for us, we played really well. In the last few weeks we have been trying to get off the bottom so it's nice to move up not just one place but two."
Meanwhile, Sons defender Martin McNiff has joined Third Division side Annan Athletic on loan for the rest of the season, linking up with former manager Jim Chapman.
Defeat for Hamilton leaves them just three points ahead of Dumbarton in the standings and gaffer Billy Reid had no complaints after a below-par performance.
He said: "We deserved nothing. It was only when we went 2-0 down that we had a bit of fight about us. Too many players didn't play to their potential."
There was some good news for Accies over the weekend, however, with keeper Kevin Cuthbert penning an extension to his contract that will keep him at New Douglas Park until May 2014.