Albion have now won just one of their last 12 games against the Wasps and were soundly beaten here – despite the apparent closeness of the scoreline.
Andy Brown should have put the visitors ahead after just seven minutes, but home keeper Myles Hogarth made a super save to deny the giant striker from close range.
The opener was merely delayed, though, and after 23 minutes Robert Sloan slammed home a penalty after Andy Graham had carelessly tripped Ross Hamilton in the area.
Graham and David Turnbull came close with headers for Stirling before the decisive second goal after 58 minutes. A mistake by Ross Forsyth allowed Richard Walker to set up David Greenhill - his shot seemed to have been kept out by Hogarth, but the linesman ruled that the ball had already crossed the line.
Hamilton and Andy Brown missed simple chances to add to Alloa's lead before Graham pulled one back in injury time, ramming the ball home from eight yards following a corner for Stirling's first league goal of the season.
It was the merest crumb of comfort for Binos boss Allan Moore, who said: "Alloa were the hungrier side and deserved to win. I can hardly recall one of my players putting in a tackle all day and I can only apologise to the fans – it just wasn't good enough."