The hosts looked set to heap further pressure on the Irishman after Barry Nicholson smashed home a spectacular opener after 22 minutes.
But Craig ensured Hibs would secure their first win since beating Dundee 1-0 on May 18 with his second-half brace.
Fenlon will be able to breathe a sigh of relief but his Killie counterpart Allan Johnston will now wonder where his side will get a win from after the defeat left him without a victory from four games in charge.
Regardless of the result, Fenlon went into the game knowing he had chairman Rod Petrie's backing after he issued a club statement on Wednesday night offering the manager a vote of confidence.
The press release was dispatched in spite of the four straight defeats which opened Hibs' season - including the 9-0 aggregate defeat to Malmo in the Europa League qualifiers.
But the early stages at Rugby Park did appear to show a hint of belief returning to the Leith side.
Rowan Vine had the ball in the net after six minutes as he slid in from Owain Tudur Jones' low cross but the goal was chalked off for offside, while the visitors strutted forward with intent on a couple of other occasions, only to be let down by the final ball.
Kilmarnock though were equally determined to make up for their winless start to the campaign, with Rabiu Ibrahim's clever feet looking promising in the early stages.
It was Nicholson's corner after 18 minutes which provided their first opening as Darren Barr looped a header just over.
But with four minutes of that chance, Nicholson showed there was real reason for Killie's self-assured display with a stunning opener.
The former Rangers, Dunfermline and Aberdeen player did not even have to break stride as he walked on to Mark Stewart's knock down from Jackson Irvine's long ball forward before volleying into the postage-stamp corner from 20 yards out.
Heffernan, however, then missed a chance to double the hosts' advantage as he deflected Sean Clohessy's cross wide of target from six yards out.
Having appeared to have lost faith in themselves following a bright start, Hibs finally clicked into gear 90 seconds after the half-time restart.
Craig gave Barr the slip as he peeled off towards the back post and Vine was pin-point with his cross from the right for the midfielder to nod into the bottom corner.
But Killie continued to threaten and first saw Mark Stewart's shot saved by Ben Williams before Barr curled a decent effort over the top of the Leith number one's crossbar.
Owain Tudur Jones came into the Hibs team in place of the suspended Kevin Thomson and offered an attacking threat of his own, seeing a low drive turned away by home keeper Craig Samson.
But it was the scorer of the Edinburgh men's first who came up with the goal that Fenlon must hope will finally kick start their season.
A long ball from Lewis Stevenson should have been dealt with by Barr but he allowed Craig to squeeze in front. Samson darted from his line in a vain attempt to clear up his team-mate's mistake but the former St Johnstone midfielder got to the ball first and lobbed the ball over the keeper and into the net.
The goal was greeted by a huge roar from the relieved Hibs supporters and Craig's jubilant celebrations cost him a booking.
Killie substitute Willie Gros fired into the side netting near the end while fellow replacement Chris Johnston wasted another glorious opening as the hosts were booed off by their disgruntled supporters.