Berwick started the game full of promise and almost opened the scoring after three minutes when a Gareth Hutchison drive was saved by Willie McCulloch.
The home side struck back, and Gary O'Connor brought off a fine save from Murphy after a quickly-taken John Paul McBride free-kick. The Borderers had the best chance of the game in the 22nd minute when Sean Paliczka beat the offside trap and bore down on goal, but then hit his shot well wide of the target.
The Warriors became stronger in the latter stages of the half, and with a minute left Tommy Sinclair thundered a shot off the crossbar.
Just before the half-time whistle blew, the Warriors took the lead after an Andy Gibson corner was only cleared to the edge of the box where Sinclair played the ball back in and Murphy powered a header goalwards. He was thwarted by a one-handed save by O'Connor, but followed up the rebound to head home.
At the start of the second half Berwick rung the changes, and for the initial stages it looked as if they would work with Palickza firing a swerving shot that was tipped over by McCulloch.
As the game progressed, however, the Warriors began to dictate the pace of the game and looked more comfortable.
Stenhousemuir looked the more likely to score, but Berwick almost equalised with four minutes left when the home defence failed to cope with a bouncing ball and Hutchison hooked a shot on to the roof of the net from close range.