Hibs have yet to score in 2011 and have now gone 11 games without a victory.
In an untidy opening to the second half, Hibs went two ahead in 55 minutes.
However, television replays showed that Zaliukas had made little, if any, contact with the Hibs man.
The 26-year-old midfielder was recently released by Port Vale and was invited to train with Hibs.
Kenny Miller tumbled when there appeared to be little contact with Hibs defender Sol Bamba.
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It was only Thomson's second goal for the Ibrox side since his arrival from Hibs.
But Hibs held firm and John Hughes in defence was inspirational in clearing any potential danger.
"All credit to Boruc and the whole Celtic defence, they defended well, " added the Hibs boss.
Mr Griffiths is in his second season-long loan to Hibs from English Championship club Wolverhampton Wanderers.
York managing director Jason McGill was firm in his rejection of the Hibs offer.
Hibs then had David Wotherspoon to thank for booting a Kyle effort off the line.
Hibs said the player realised that "despite the provocation", his language had been "unacceptable".
The 22-year-old, who joined Hibs from Sunderland last summer, has three caps for the Republic of Ireland.
Icelander Victor Palsson sent Jaakkola the wrong way from the spot scoring his first goal for Hibs.
Hibs had obviously not heeded the warning as Templeton netted a stunning solo goal three minutes later.
The Falkirk defence were all at sea and it was no surprise when Hibs went further ahead.
Paul Hanlon restored Hibs' advantage from close range before Higdon drove home to signal a rousing finish.
Hibs have slipped to 10th in the Scottish Premier League, just five points above bottom club Hamilton.
Hibs move up to second place in the Scottish Premier League, leapfrogging Rangers, who play on Sunday.
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Kenny Deuchar's header increased the home side's lead before Anthony Stokes pulled a goal back for Hibs.
And Goodwillie made Hibs pay for their profligacy, latching onto Danny Swanson's long ball to fire home.
Sutton came off the bench to net an 89th minute equaliser in Wednesday's 1-1 draw with Hibs.
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Graham Stack made several important saves to keep Hibs in the match early in the second half.
Hibs finally found a way into the Kilmarnock box in the closing minutes of the first half.
Hughes took heart from some of his side's performances and thought the best player was in Hibs' midfield.
Hibs manager John Hughes has replaced Stokes by agreeing a loan deal with Bristol Rovers for Darryl Duffy.
However, Hibs, who were dismal in a drab first half, responded well with debutant Ricardo Vaz Te lively.
Hibs can no longer catch third-placed Motherwell, who visit Easter Road on the final day of the season.
Just when United were beginning to look comfortable, Hibs took the lead through Fletcher after a great move.
Hibs were looking to bounce back from last weekend's 3-0 defeat by Rangers and dominated the early possession.
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