David Hopkirk put Stenny in front for the first time with his fourth goal this season midway through the first half and former Celtic youth player Mark McGuigan's penalty on the stroke of half-time extended their lead.
Martin Scott's late header reduced the deficit but Stenhousemuir held on for the win to extend their unbeaten run to three matches as they moved up a spot to fourth, while Cove's three-point lead at the top was reduced to one as a result.
Moments later, the away side lost captain Craig Thomson to a straight red card for a bad challenge as they dropped down to third.
Annan advanced to fifth after returning to winning ways with a 2-1 victory away to Queen's Park, who have lost their last three.
Annan summer signing Scott McLean opened the scoring against his former side after 18 minutes and – four minutes later – Tommy Muir doubled their lead before David Galt pulled one back six minutes from time.
Elgin moved up two places to sixth thanks to a 3-1 win away to Albion Rovers.
Elsewhere, the bottom two sides Stirling and Brechin shared the spoils from a 1-1 draw at Glebe Park.
Brechin top scorer Paul McManus netted his second goal in two games as his 47th-minute header broke the deadlock but former Brechin defender Paul McLean, who left Glebe Park after a decade this summer, headed the visitors level midway through the second half.