Winger Fati, who is still 16 years old, bagged a goal and an assist for Frenkie de Jong in the first seven minutes of his full debut for the club, having become its youngest league goalscorer as a substitute in the draw with Osasuna last month.
Kevin Gameiro pulled one back for Valencia, which caused a shock this week when it sacked head coach Marcelino and replaced him with the inexperienced Albert Celades, but Gerard Pique effectively killed off the contest 51 minutes in.
Luis Suarez scored his first goal of the season with his second touch after coming on as a substitute and added a late second to seal the win, as Lionel Messi — still not risked as he recovers from a calf injury — watched on approvingly. Maxi Gomez pinched a late consolation, his first Valencia goal.
In just the second minute, Fati met De Jong's cut-back and rifled home his first Camp Nou goal for the club.
He turned provider five minutes later, dancing his way into the box from the left and teeing up De Jong to open his account for the club with a powerful side-foot finish.
Barca appeared in total control but allowed Valencia a lifeline when Gameiro sprung the offside trap and slotted in off the left-hand post, a VAR (video assistant referee) review granting him a reprieve after the goal was initially disallowed.
Ernesto Valverde's side restored Barcelona's two-goal lead six minutes after the interval, as Pique pounced to tap home the rebound after Jasper Cillessen could only parry an Antoine Griezmann shot against the woodwork.
Nelson Semedo then rattled the other post as Suarez came on for Fati, and the Uruguay star promptly scored with his second touch of the ball, feinting before bending a shot into the bottom-left corner.
Suarez hit almost the same spot with eight minutes to go for Barca, further highlighting the difficult task awaiting Celades over the coming weeks.
Barca begins its UEFA Champions League duties away to Borussia Dortmund on Wednesday (AEST), with a league trip to Granada to follow four days later. Valencia heads to Chelsea before a home league meeting with Leganes.