The Welshman, who has been in charge for two years, took Barnsley to a Wembley FA Cup semi-final last season, but faces a showdown with club owner Patrick Cryne this week.
Davey's hopes of surviving the axe were given a huge shot in the arm as they chalked up only their second win of the season.
Victory came with a spirited second-half fight back after they were hit by a first-half penalty and 49th minute red card for defender Darren Moore.
Disaster struck for the home side in the tenth minute when Rob Kozluk's push on Lewis Guy in the area earned the visitors a penalty.
Brian Stock slotted home the spot-kick out of the reach of Heinz Muller despite the keeper's full-stretch dive to his right.
But Muller produced a super save to deny Rovers a second when Martin Woods spun superbly on the edge of the area and unleashed a goalbound left-foot shot.
And when Stock took the corner, Matthew Mills ghosted in for a free header that was straight at the keeper.
Kozluk came close to making amends for the penalty when he darted forward and drilled a low angled shot that bounced just in front of Neil Sullivan, but the Rovers keeper turned the chance away.
Moore was dismissed when he dragged down Guy but it sparked an incredible response with two goals in six minutes.
First Denis Souza headed on a cross and Stephen Foster connected with a header to beat Sullivan.
Jamal Campbell-Ryce, who turned on the second-half style, then raced in and rifled home the second with superb angled shot.
And in the 73rd minute, John Macken crowned the victory when he followed up after Sullivan saved Martin Devaney's shot.
Three minutes from the end, Iain Hume rounded off the scoring when he smashed in a shot from the edge of the box.