The early stages were dominated by the swirling rain and greasy surface as neither side appeared capable of passing the ball to a team-mate.
The only highlights of the opening 20 minutes were three bookings in the space of two minutes, with Brian Kerr and Stephen Craigan seeing yellow for Well and Kestutis Ivaskevicius being booked for Hearts.
In the 24th minute Well should have taken the lead when Jamie Paterson sprang the offside trap from Danny Murphy's throughball. He rounded Craig Gordon only to leave himself too much to do and he eventually ran the ball out for a goal kick.
Hearts looked the livelier side with Andrew Driver twice going close to breaking the deadlock with dipping shots.
The visitors were eventually rewarded for their enterprise with the opening goal eight minutes before the break. Once again Driver started it all off when his shot was blocked by Graeme Smith at the near post. From Laryea Kingston's corner Ibrahim Tall rose unmarked to bullet a header into the net.
Well only managed one real effort at goal in the first half. In the 40th minute Smith ran on to a David Clarkson flick but he could only shoot tamely at Gordon from 15 yards out.
Both sides had early second-half chances with Michal Pospisil shooting wide for Hearts and Krisztian Vadocz volleying straight at Gordon at the other end.
The Steelmen should have equalised in the 58th minute when Kingston was forced to clear a Clarkson header off the line.
The next goal was always going to be crucial and the visitors grabbed it against the run of play in the 66th minute when Well defender Craigan met Kingston's cross and contrived to head the ball past his own keeper at the near post.
Well almost snatched a late consolation goal but substitute Tomas Kancelskis recovered to block Clarkson's effort away for a corner.
Minutes later Trevor Molloy could have capitalised on a rare Gordon fumble but was caught on his heels and Hearts scrambled the ball clear.
In the dying minutes Kingston once again kept his side two up with another goal-line clearance, this time from Mark Reynolds' shot following Paterson's corner.
Ross McCormack was lucky not to be sent off in the closing stages after a reckless challenge on Gordon as the keeper attempted to gather the ball on the edge of the box.