The winger's 34th-minute cracker, coupled with an array of superb saves from goalkeeper Jamie MacDonald, inspired the Jambos to their first victory in 10 games.
Hibernian dominated for large spells and created a host of clear-cut chances but they found MacDonald in unbeatable form, and their hopes of an equaliser were all-but ended when skipper James McPake was dismissed with six minutes remaining.
Such a result seemed highly unlikely during the opening exchanges, where Hibs were utterly dominant. Scott Robertson was causing problems on the right flank and he was the creator when Paul Hanlon saw a fifth-minute header come back off the post.
Robertson then forced MacDonald into two crucial saves in three minutes, the second of those a wondrous effort that deflected the midfielder's shot onto the crossbar.
James Collins and Hanlon again went close for the hosts but they were rocked on 33 minutes when Stevenson scored from Hearts' first attempt on target.
The 29-year-old, who returned to the starting line-up after shaking off a knee problem, was played into space by Jason Holt and rifled an unstoppable shot past Ben Williams.
Stevenson's goal rocked Hibs and they struggled to regain any momentum either side of the interval, and tame efforts from Collins and Rowan Vine did little to quell the impatient home support.
Despite failing to hit the heights of their first-half performance, Hibs should have equalised with 16 minutes remaining when Vine played Liam Craig through on goal, but he shot straight at MacDonald.
A miserable night for Pat Fenlon's side was complete on 84 minutes when McPake received a straight red card for an ugly two-footed lunge on Callum Paterson, as Hearts held on for a morale-boosting victory.