Hearts lead Rangers by four points with 10 matches to go and enjoy a six-point cushion over Celtic, who have a game in hand.
No side outside the Old Firm giants has won the title in 41 years.
A first league defeat in two interim spells in charge of Celtic for Martin O'Neill this season rounded off a miserable few days for the defending champions.
O'Neill's men were thrashed 4-1 by Stuttgart in the Europa League on Thursday amid fan protests over a lack of investment in the side in recent transfer windows.
A seventh league defeat of the campaign leaves the Hoops with a mountain to climb, especially with a visit to Rangers to come next weekend.
Felix Passlack fired Hibs in front, but Benjamin Nygren levelled before half-time.
Celtic have made a habit of late winners in recent weeks, but were forced to play out the final 16 minutes a man down when Auston Trusty was shown a red card after a VAR review for an-off-the-ball incident.
Hibs made the man advantage count when Kai Andrews blasted in the winner in the 87th minute to do their Edinburgh rivals a huge favour in the title race.
Livingston have not won in 26 league games but were minutes away from stunning Rangers and did enough to inflict a blow on their title chances.
Rangers trailed 2-0 inside the final 10 minutes as goals from Brooklyn Kabongolo and Lewis Smith gave Livie the lead.
However, a red card for Cristian Montano on the hour mark proved to be the turning point.
Emmanuel Fernandez reduced the deficit with a thunderous strike in the 81st minute and Mikey Moore levelled with two minutes remaining but Marvin Bartley's side held out during nine added minutes to clinch a point.