The opening period was a poor affair but both teams should at least have scored.
Alloa's Paul McLeod lobbed over from the edge of the box after Dumbarton keeper Stephen Grindlay had saved a rebound from one of his own defenders while at the other end Iain Russell somehow managed to fire wide from a handful of yards following a great cross by John Dillon.
After the break it was Alloa who were on top and they were awarded a free-kick on the Dumbarton six-yard line after Grindlay had collected an alleged pass-back, but Stevenson's effort was blocked by the mass of yellow jerseys on the goal-line.
He was not do be outdone, however, and after 76 minutes Robert Sloan picked him out at the back post and Stevenson's header went into the net via the underside of the bar.
Dumbarton left the field to chorus of boos and under-fire manager Paul Martin will have to turn things round extremely quickly.