Livingston weathered the early Hibs whirlwind, in which Rocky Bushiri had a smart volley ruled out for offside, and hit back with a beautiful goal of their own.
Building from back to front, a series of swift passes ended with Scott Pittman pulling the ball back for Jeremy Bokila to apply the deftest of finishes into the bottom corner as parity was restored before the interval.
Pittman also felt unlucky to have a goal of his own disallowed earlier, when Callum Scott penalised Bokila for a foul on Grant Hanley in the build-up.
Hibs had beaten Livingston twice this season, with an aggregate score of 6-0, and looked like they would add a third triumph.
Restored to the starting XI after being left out against Rangers in midweek, Martin Boyle repaid David Gray's faith with a fine touch and turn inside the box to give himself the easiest of tasks to bury the ball beyond Jerome Prior.
Livingston's late leveller was just, and nearly came earlier than Carey's well-taken penalty. After a scramble in the Hibs box, substitute Tete Yengi should have scored but his weak effort was saved by keeper Jerome Sallinger's legs.
David Martindale's men refused to accept defeat, though, and deservedly drew level with virtually the last kick of the ball. They stay bottom - and are still looking for their first win since 9 August - but do edge closer to Dundee.
As for Hibs, spurning two points late on will sting, but they still move back into third place, ahead of Rangers and Falkirk on goal difference.