Giovanni van Bronckhorst’s side, who were taken to extra-time before eventually overcoming Braga to reach the Europa League semi-final stage on Thursday night, defied further energy sapping exertions to book a final showdown against Hearts next month.
Celtic defender Greg Taylor opened the scoring after 64 minutes in a pulsating contest before substitutes Scott Arfield and Sakala completed a stunning comeback victory.
The national stadium was bursting at the seams as the players emerged, with both sets of supporters letting of smoke bombs and flares which shrouded the pitch andmade visibility difficult prior to kick-off.
This was part three of an Old Firm trilogy over a six-week period as Celtic aimed to keep alive their dream of a fifth Treble in six years as they looked to build on impressive recent momentum.
Given the Hoops were in complete disarray 12 months ago Parkhead boss Ange Postecoglou has totally transformed the club’s fortunes.
However, the Australian was without in-form striker Giorgos Giakoumakis due to a hamstring injury sustained against St Johnstone last weekend, paving the way for Liel Abada to return to the starting line-up.
Light Blues manager Giovanni van Bronckhorst made just one change from the side that knocked out Braga 3-2 on aggregate to set up a Europa League last-four tie against RB Leipzig.
First-choice goalkeeper Allan McGregor dropped out with Jon McLaughlin taking his place between the sticks.
Despite coming into this clash six points adrift of Celtic, Rangers will not give up their Premiership title defence until it’s mathematically possible judging by this heroic performance.
The Scottish Cup represents their best opportunity to lift domestic silverware this season but they haven’t etched their name on the famous trophy since 2009.
The pre-match build up centered around Rangers energy levels after they were taken the distance by the Portuguese outfit just three days earlier.
However, there was no need for Van Bronckhorst to lift his fatigued players psychologically for a game of this magnitude.
The Dutchman dispeled any notion of tiredness in his press conference on Friday and his players certainly flew out the traps in front of an electric atmosphere.
John Lundstram won back possession before picking out Aaron Ramsey on the edge of the box but his pass was just behind Ryan Kent who was unable to get a clean shot away.
Celtic came into the game in formidable form and were chasing a third consecutive derby victory, having narrowly prevailed 2-1 at Ibrox a fortnight ago.
Full-back Josip Juranovic delivered a perfectly weighted cross for Jota at the back post but the on-loan Benfica winger nodded just wide of the target in a breathless start to the match.
The high-tempo nature of the game continued as Kemar Roofe dragged an effort narrowly wide of the post before Lundstram curled a 20-yard strike off the crossbar after 40 minutes.
Rangers were preventing their bitter rivals from playing the free-flowing football Celtic fans have become accustomed to seeing under Postecoglou’s reign.
Their excellent positional play limited the influence of Celtic’s creative midfielders. That was until Aaron Ramsey was forced off injured after coming off worse in a challenge with Cameron Carter-Vickers.
Jon McLaughlin had virtually been a spectator during the first-half as his team mates continued their search for an opener earlier in the second period.
Roofe was fortunate to escape with just a caution after flying into a challenge with Daizen Maeda before Ryan Kent cut inside from the left but his effort flashed inches wide of Joe Hart’s goal.
That sparked Ange Postecoglou into a double substitution just before the hour mark as Kyogo Furuhashi and Matt O’Riley replaced Abada and Rogic.
Those changes instantly made an impact and the deadlock was broken on 64 minutes somewhat against the run of play.
Rangers switched off at a quickly taken free-kick, allowing skipper Callum McGregor to find Greg Taylor and the left-back’s shot deflected off Calvin Bassey and wrong-footed McLaughlin into the corner of the net.
Carter-Vicker then crashed an effort off the bar after Rangers failed to clear a corner.
Just when it appeared as though the Hoops were beginning to dominate, Rangers responded emphatically as Arfield hauled them back on level terms after 78 minutes.
The substitute took the ball of Roofe inside the penalty area before bending an effort beyond Hart.
That goal gave the Gers a massive lift and Roofe had the ball in the net minutes later, only for the Jamacian to be correctly flagged offside.
Celtic will have been glad to hear the full-time whistle and force an extra 30 minutes as they soaked up a lot of late pressure.
Rangers were now having to call on all their remaining reserves of energy as both teams continued to slog it outon a warm afternoon.
The woodwork came to Celtic’s rescue after 110 minutes when James Tavernier’s angled shot was pushed onto the post by Hart. The ball then looped up for Fashion Sakala but his overhead kick was pushed over the bar by the Celtic No.1.
The Zambian was not to be denied moments later as he latched on to Bassey’s cut-back before turning the ball home via a deflection of Carl Starfelt.
It was testament to Rangers outstanding fitness levels that they managed to cling on and book a return trip the Hampden next month.
It capped off a memorable week for Van Bronckhorst and his players as they celebrated a deserved victory built on desire and real hunger.