Rangers continued to dominate proceedings as the first half progressed and were unlucky on two separate occasions towards the end of the period.
First midfielder Arnold Peralta saw his ambitious shot from 40 yards come back off the frame of the Stenhousemuir goal and then Nicky Law was denied by the crossbar when he tried his luck from 25 yards.
That miss came back to haunt the Warriors as Rangers found the net in the 74th minute. Goalkeeper Chris Smith came rushing out to collect a long ball but failed to grab it, allowing Daly to head home from 16 yards.
Law came close to making it 2-0 in the final 10 minutes of the encounter when he curled a shot beyond Smith, but Kevin McKinlay made a good goal-line clearance.
Then in the dying moments of the contest, Lee McCulloch was denied by the frame of the Stenhousemuir goal when his header from a corner crashed against the crossbar.
That did not matter in the end, though, as Rangers marched on to the final of the Ramsdens Cup where they will take on Raith Rovers.