The Bluebirds had trailed when Robert Snodgrass gave Norwich the lead in just the fifth minute at Cardiff City Stadium.
But substitute Zaha teed up Craig Bellamy for the equaliser just after the break, with Jones firing home the winner at the second attempt just over a minute later.
It ensured Ole Gunnar Solskjaer secured a first Premier League win since taking charge of the Welsh club which, while not moving them out of the drop zone, got them off the bottom of the table at Fulham's expense.
And the performance of Zaha, whose introduction before half-time suddenly enlivened a Cardiff side desperately lacking a spark, will give Bluebirds fans real hope of surviving the drop.
It was Cardiff's first win in eight league games, and Norwich have now won just one of nine as they remain firmly in the relegation battle having wasted countless chances in a game they fully deserved something from.
The arrivals of Fabio, Jones and Zaha in the week had buoyed the Bluebirds faithful, and within 60 seconds Fabio had linked well with Bellamy and Craig Noone in a move which ended with Jordon Mutch failing to connect with an ambitious overhead kick.
But Fabio's next contribution proved nowhere near as positive as his first.
The full-back was outflanked by Martin Olsson despite Bradley Johnson's pass into the channel being so telegraphed everyone else in the ground could see it coming.
Olsson played the ball across the box where Snodgrass had the easy task of finishing from eight yards.
A woefully misplaced backpass from Fabio almost gave Gary Hooper sight of goal seconds later, and Norwich should really have made more of a spell where Cardiff were all at sea defensively.
Hooper failed to get his head on the end of Russell Martin's excellent cross, before Johnson blazed over after Johan Elmander had easily held off Ben Turner.
Cardiff slowly established a foothold, although they failed to work John Ruddy in the Norwich goal during the opening 45 minutes.
Noone whistled an effort wide before producing a wonderful pass which Mutch failed to connect with.
Solskjaer had seen enough and withdrew Peter Whittingam in just the 38th minute, seconds after the midfielder had dragged an effort narrowly wide, to hand Zaha his debut.
The anxiety was tangible at the interval, but the game turned on its head within minutes of the resumption.
Hooper's ball across his own box was seized upon by Zaha, who found Bellamy's angled run for the striker to become the first player to score for seven Premier League clubs.
Just 73 seconds later, Cardiff led.
Jones, largely ineffectual during the first half, lashed home at the second attempt after Martin had blocked his initial header from Noone's cross.
A previously drab contest was suddenly transformed and Johnson thought he was about to equalise for Norwich, only for David Marshall to tip his lofted strike onto the bar.
Cardiff were next to strike the woodwork. Gary Medel won the ball on the edge of the Norwich box and squared to Mutch. The midfielder's sliced shot was going wide but struck Jones on the chest and rolled agonisingly against the post.
Another lovely pass from Zaha saw Bellamy go tumbling as he raced into the box, but referee Mark Clattenburg waved away penalty appeals.
Desperate for victory, Cardiff dropped ever deeper during the final quarter, and needed heroics from Marshall to keep them in front.
The Scotland international saved low to his right from Nathan Redmond, who also had an effort ruled out for offside, before Bellamy headed Hooper's effort off the line.
Substitute Leroy Fer worked Marshall again from point-blank range , and Norwich's final chance came and went when Snodgrass headed against the post from a Redmond corner.