Josh Campbell gave Hibs the lead early in the game from the penalty spot before Miguel Chaiwa turned a Gassama ball into his own net to draw the game level in the 84th minute.
However, it was the visitors who found the winning goal in injury time through Callum Wright.
There were six changes from Thursday’s UEFA Europa League qualifier in Poland, including a full debut for Olwethu Makhanya, while Vanja Dragojević made his first start for the club.
Ivor Pandur once again started between the sticks, with defender Makhanya debuting in a backline alongside Emmanuel Fernandez, Tuur Rommens and Dragojević.
Cammy Devlin and Nico Raskin were in midfield, while wingers Findlay Curtis and Daisuke Yokota were both named from the start alongside Lawrence Shankland and Youssef Chermiti.
The atmosphere was electric ahead of kick-off as a sold-out Ibrox prepared for the first competitive home match of the season. But with less than 10 minutes on the clock, it was the visitors who were awarded a penalty and the opportunity to take the lead after a VAR check. Dragojević was judged to have fouled Jason Kerr inside the area, as both players battled for the ball.
Josh Campbell stepped up to the spot and sent Ivor Pandur the wrong way with his right-footed penalty to give the visitors an early lead.
Gers searched for an instant response as Chermiti got on the end of a long ball forward from Fernandez but, from a tight angle, sent it wide, before Cammy Devlin had an effort held comfortably by Sallinger.
McInnes’ men were probing for an equaliser as the game passed the 20-minute mark; Curtis set up a strike for Vanja Dragojević on the edge of the box, but his right-footed hit curved just wide of the post.
The best chance so far came shortly after; it was a fantastic bit of play out on the right from Daisuke Yokota as he beat the defender at the byline to cut back for Chermiti. The Rangers number nine got a good connection on the shot, but Megwa cleared off the line.
Tuur Rommens was next to force a save from the goalkeeper as his left-footed strike from distance was pushed away. In the aftermath, Chermiti tried to keep the ball in play at the byline and was injured in the process with what looked like a serious one as he was stretchered off the pitch, with Ryan Naderi coming on to replace him.
Before the break, Dragojević flashed a looping strike just wide from distance, before Fernandez lifted a nice ball over the defence and into the path of Raskin, but he couldn’t keep his effort down.
Hibs had a chance in injury time as they broke forward in numbers, with Josh Campbell’s shot blocked by Fernandez.
McInnes made one change at the break as Mohamed Diomande was introduced for Vanja Dragojević, with Findlay Curtis stepping back into right-back.
Curtis saw a shot blocked in the 53rd minute, with Diomande’s rebound pushed away by Sallinger. Raskin came close with a low drilled shot which was saved by the goalkeeper as Gers continued to pressure.
Shankland thought he had his first Rangers goal with just over an hour played as the ball broke for him on the edge of the box. Sallinger got a touch on the shot to tip it onto the crossbar before the danger was cleared.
Another double change followed, this time Djeidi Gassama and Thelo Aasgaard were introduced to freshen up the attack, in place of Nico Raskin and Daisuke Yokota.
With 84 minutes on the clock, Rangers had the goal they had craved. Aasgaard picked up on a loose ball and found Gassama. He sent the ball into the box, and Miguel Chaiwa got the final touch to send it into the back of his own net.
It gave Gers and the home crowd a lift, and Shankland came close to a goal as his header was hooked away by Sallinger. Gassama threatened in injury time with a strike from distance which was blocked by the goalkeeper’s feet.
Despite Rangers' pressure, it was Hibs who got the injury-time winner as they broke up the pitch. Callum Wright cut inside and sent a low right-footed shot towards goal which sneaked into the bottom corner.
RANGERS: Pandur, Makhanya, Fernandez, Rommens, Dragojevic (Diomande, 46), Devlin, Raskin (Aasgaard, 68), Yokota (Gassama, 68), Shankland, Curtis (McCrorie, 79), Chermiti (Naderi, 35).
Subs: Kelly, Sterling, Neil, Chukwuani.
Hibernian: Sallinger, Mayor (Elding, 90), O’Hora, Kerr, Mulligan, Chaiwa, Klidje (Lowe, 58), Obita, Suto (McGrath, 58), Megwa (Passlack, 76), Campbell (Wright, 58).
Subs: Smith, Newell, Allan-Molotnikov, Abduaai.