The Catalan giant had won just three of its opening seven LaLiga games before the visit of Los Che and more disappointment looked a real possibility when Jose Gaya powered Jose Bordalas’ side ahead early on.
Fati, whose return from a long-term knee injury had been limited to substitute appearances before the international break, marked his first start since 7 November with a superb equaliser.
Depay put the Blaugrana in front from the penalty spot before the interval.
Substitute Philippe Coutinho gave the scoreline a healthier look five minutes from full-time as Barca, which handed former Manchester City striker Sergio Aguero a late debut from the bench, sealed an important win.
It took just five minutes for Valencia to take the lead, Gaya lashing a superb half-volley into Marc-Andre ter Stegen's bottom-right corner from 30 yards.
Barca responded well, though, and drew level eight minutes later when Fati whipped past Jasper Cillessen from just outside the penalty area after a one-two with Depay.
The Dutchman then put the hosts ahead four minutes before the interval, thumping home from the penalty spot after Gaya had brought down Fati inside the area.
Cillessen got down well at his near post to keep out Fati's effort shortly after the restart, while at the other end Carlos Soler saw an effort crash back off Ter Stegen's right-hand post.
Ter Stegen had to be alert to keep out Goncalo Guedes' powerful strike shortly before the hour mark as Valencia threatened to restore parity.
Maxi Gomez almost drew Valencia level with a long-range free-kick that flashed past Ter Stegen's upright inside the final 10 minutes, yet it was Coutinho who had the final say, the Brazil international sliding in his first goal of the season from close range.