Arsenal had work to do after leaving Spain with a 2-0 deficit last week and they came out full of intent at Emirates Stadium, knowing what was at stake.
Their season has not gone to plan, but boss Slegers has got them back on track in this competition and this will go down as her most significant victory since taking over in January.
England international Kelly was a constant threat, whipping teasing balls into the box and running in behind Real Madrid's backline.
The Gunners had been completely dominant in the first half, having 26 touches in the visitors' box and six shots - but Kelly's inviting crosses had gone untouched.
Their best chance had fallen to Caldentey and she should have scored when left completely unmarked at a corner, but headed the ball on to the crossbar.
But Arsenal had nothing to show for their efforts and momentum had swung in Real Madrid's favour when the teams headed down the tunnel at half-time.
However, the game - and the home side's fortunes - transformed within 43 seconds of the second period.
Buoyed by an optimistic home crowd and their strong first-half display, Arsenal were rewarded for a blistering response.
Kelly delivered one exquisite delivery after another, teeing up Russo for her crucial opener, then Caldentey's header which doubled Arsenal's lead within three explosive minutes.
A third goal seemed inevitable as everything they touched turned to gold and supporters got their money's worth when Russo struck again to cap off an incredible turnaround.
The England forward deserved a hat-trick and had three opportunities to get one – VAR preventing two of them for marginal offsides and visiting goalkeeper Rodriguez resolute to block well as the forward raced through midway through the second half.
But Russo will sleep soundly knowing Arsenal's European hopes remain intact - and she played a major part with a fine frontline performance, taking her Champions League tally this season to six goals.
It has been 18 years since they last reached the final of this competition - when they won a historic quadruple - but they can start believing again as they prepare to face record holders Lyon next.