There was nervousness about Rangers in the early stages and Hibs found the opener through Josh Campbell's 12th-minute penalty, which was awarded by VAR for a foul by Vanja Dragojevic on Jason Kerr.
Rangers, who had 25 shots, got level in the 85th minute when Djeidi Gassama's low cross was turned in by Hibs' Miguel Chaiwa.
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But there was a late twist to come as Hibernian broke from a corner and Callum Wright's deflected shot found a way into the far corner.
It capped a miserable first game for McIness at Ibrox, who also saw star striker Youssef Chermiti - subject to a failed bid by Galatasaray for a reported club record fee of £22m - carried off the pitch on a stretcher.
With Celtic and St Mirren both on six points from two Premiership games, Rangers are five points behind and already chasing with McInnes looking for his first win after three matches in the Ibrox hot seat.
The key moments from Ibrox
12: VAR! The referee is instructed to come to the screen to review a grapple between Dragojevic and Kerr. He awards the penalty.
13: GOAL! Campbell buries the spot kick to give Hibs a 1-0 lead.
25: OFF THE LINE! Chermiti thinks he's scored but Megwa makes a brilliant clearance off the line.
37: CHERMITI STRETCHED OFF! The Rangers striker takes a nasty tumble into the advertising boards and suffers a match-ending injury.
62: OFF THE BAR! Lawrence Shankland's deflected effort is tipping onto the crossbar by Raphael Sallinger.
83: GOAL! Rangers finally make the breakthrough as Chaiwa's turns the ball into his own net.
90+3: GOAL! With Rangers pushing for the winner, Hibs break and Wright manages to squeeze a shot into the bottom corner.
