Vigurs was denied by Balogh in the closing stages but Hearts held on for victory.
Offiong met Simon Mensing's back-post cross but could only head tamely into the arms of Balogh.
Hearts have also made an offer to sign Hungary goalkeeper Janos Balogh on a six-month loan deal.
Caley pushed forward with Ian Black finding Dougie Imrie, but he rushed his effort and Janos Balogh gathered easily.
Motherwell forced two late corners but Mark Reynolds' header was straight at Balogh at the death as Hearts held on.
Laszlo also revealed Janos Balogh could retain his place in goal despite Marian Kello resuming training following a head injury.
Dr Ruth Balogh, nuclear issues campaigner for West Cumbria and North Lakes Friends of the Earth, said she was "absolutely delighted".
Kello replaced the injured Janos Balogh in goal and Lee Wallace and David Obua were also restored to the starting line-up.
Forbes also blazed over with a free-kick from 14 yards after Balogh had been penalised for picking up a Jose Goncalves back-pass.
He becomes Hearts' third signing of the summer and the move will almost certainly end Hearts' interest the Hungary goalkeeper Janos Balogh.
Fox flighted in for the Dutchman to snatch the win after Balogh had flapped a feeble clearance inside the six yard area.
BBC: SPORT | Football | Scottish Premier | Celtic 2-1 Hearts
As Hearts unsuccessfully appealed for Smith to allow treatment to the prostrate Mrowiec, the United striker saw his weak penalty saved by Balogh.
John Sutton's glancing header wide from a McLean cross offered an immediate response, before David Clarkson drove straight at Janos Balogh from distance.
Michael Higdon slipped a pass through to Lovell and the striker ran clear of the defence before toe-poking the ball past goalkeeper Janos Balogh.
Lee Barrowman was left disappointed as the half-time whistle approached when he stabbed the ball past Balogh, but failed to get it past the Hearts defence.
With 71 minutes played, McDonald struck the inside of Balogh's right-hand post and Killen hesitated on the re-bound before making a hash of his follow-up shot.
BBC: SPORT | Football | Scottish Premier | Celtic 2-1 Hearts
Accies almost capitalised on their one-man advantage when David Graham sent in a low drive which Janos Balogh saved before Marius Zaliukas cleared for a corner.
The visitors made the perfect start, with Invincibile meeting a Mehdi Taouil cross on eight minutes and steering a bullet header beyond Janos Balogh from close range.
Hearts goalkeeper Janos Balogh looked uncertain when dealing with corners and he had to tip over Shaun Hutchinson's header after Sutton had headed O'Brien's delivery back across goal.
Hearts goalkeeper Janos Balogh denied Steven Fletcher a close-range winner.
Ari Balogh, current head of products and technology, will be leaving the company on June 3 for personal reasons and will work closely with Irving to ensure a smooth transition.
The German right-back's low cross was volleyed first time by Vennegoor of Hesselink and goalkeeper Janos Balogh was slow to get down to his left as the ball trickled into the net.
Hearts were missing as an attacking force, but they comfortably survived during a scrappy ending to the half, which saw Warren Feeney booked by referee Eddie Smith for a late challenge on Balogh.
The visitors lost their skipper to a nasty looking injury but were soon celebrating when a thunderous Hughes free kick flew through a ruck of bodies and eluded Balogh to find the net.
Hearts looked to be dead and buried when Balogh was sent off for chopping down Fletcher after the striker was sent clear through the middle by Colin Nish on the stroke of half time.
McGeady blasted in a dangerous left-footed shot from 25 yards out that caught out Balogh and rebounded off the bar then, 10 minutes later, swerved a shot at the Hearts goal that curled wide.
BBC: SPORT | Football | Scottish Premier | Celtic 2-1 Hearts
But Hearts saw out the remainder of the game - including five minutes of injury time - with Balogh making a comfortable save from Dods after substitute Jon Daly had laid off Caddis' clever pass.
Killen found the equaliser with a diving header in the 56th minute after his earlier long-range shot was kept out by a fully-stretched Balogh and Maloney had cut back a fine cross to the back post.
BBC: SPORT | Football | Scottish Premier | Celtic 2-1 Hearts
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