Two goals in two minutes from substitutes Tete Morente and Pere Milla drew the away side level with 25 minutes to play before Nico scored for the second game running.
Barca went into the game without a win in three matches in all competitions and with the majority of their attack injured. Ansu Fati, Memphis Depay, Pedri and Martin Braithwaite remain sidelined, while Sergio Aguero confirmed his retirement from football due to a heart condition earlier this week.
That means a first start for Jutgla, a summer signing from Espanyol for the B team. He had already had an early strike ruled out for offside when he flicked home Ousmane Dembele's corner to open the scoring.
Almost immediately, Barca added a second. A brilliant touch and turn opened up space for Gavi, who drove into the Elche box before shooting low into the bottom corner. It was the 17-year-old's first goal for the club.
Barca spent the rest of the first half in cruise control, with Abde Ezzalzouli, Jutrgla and Gavi all missing chances to make it three, although Elche defender Enzo Roco should have pulled one back when he was left unmarked at a free kick.
The visitors were not so wasteful when chances came their way in the second half, with two goals in two minutes turning the game on its head.
Lucas Boye did well to keep the ball from going out of play and Morente took full advantage of the space left by Barca's defence, taking on Clement Lenglet and firing past Marc-Andre ter Stegen in 62nd minute.
A minute later, Milla completed the comeback. He was left unmarked at the far past to bundle a header over the line after Ronald Araujo had given possession away for Barca.
Xavi brought Nico on for Jutgla and Riqui Puig for Frenkie de Jong as Barca pushed for the winner. It eventually arrived when the impressive Gavi linked with Dembele before teeing up Nico, who sealed Barca's first win since Nov. 27.