New boys Jones, Zaha lift Cardiff
Substitute Zaha set up Craig Bellamy for the equaliser after the break with Jones firing home the winner just two minutes later.
Robert Snodgrass had put Norwich ahead after five minutes.
It gave Ole Gunnar Solskjaer a first Premier League win since taking charge of the Welsh club and got them off the bottom of the table at Fulham's expense.
Zaha had started on the bench on his debut following his loan move from Manchester United, but came on minutes before the break to replace Peter Whittingham.
Brazilian full-back Fabio and Trinidadian striker Jones were also handed their Cardiff debuts, with captain Mark Hudson ruled out with a hamstring problem.
Norwich boss Chris Hughton brought in Alexander Tettey, Jonas Gutierrez and Johan Elmander.
And it was Norwich who got off the mark fastest in a tie between two of the lowest scoring teams in the Premier League.
Jonas Gutierrez picked up a Russell Martin cross after three minutes but his header from the centre of the box was high and wide.
But two minutes later Snodgrass made no mistake.
The tables were turned after the break as the game came to life.
Zaha set up Bellamy and the Welsh striker became the first player to score for seven Premier League clubs.
Jones' presence was also felt in the second half and he fired home at the second attempt after his initial header was blocked.
Johnson almost equaliseed for Norwich but David Marshall tipped his strike over the bar, and Cardiff needed some heroics from Marshall to keep them in front in the final quarter.
The Scotland international saved from Nathan Redmond, who also had an effort ruled out for offside, before Bellamy headed Hooper's effort off the line.
Norwich's final chance came when Snodgrass headed against the post from a Redmond corner, as they remain two points above the drop zone.