Winger Fati, just 16, got a goal and an assist for Frenkie de Jong in the first seven minutes of his full debut for the club, having become their youngest league goalscorer as a substitute in the draw with Osasuna last month.
1 - Ansu Fati (16 years and 318 days) has become the youngest player to both score and assist in a LaLiga game in the 21st century. Diamond. pic.twitter.com/jk2zA8ddbZ— OptaJose (@OptaJose) September 14, 2019
Kevin Gameiro pulled one back for Valencia, who caused a shock this week when they 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 following 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 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.
What does it mean? Barca take advantage of Atletico slip-up
Atleti's 2-0 defeat to Real Sociedad earlier means there is not a single team in the division with a 100 per cent record this season, despite only four rounds of matches having been played.
Barca made the most of Atleti's loss to move within two points of the early league leaders, while the manner of the performance will give Valverde plenty of encouragement - particularly ahead of the start of the Champions League.
Fati's rise shows no sign of slowing
Fati started the season pushing for a regular place in Barca's B team, making his emergence under Valverde all the more remarkable.
He showed a clinical touch for his goal and some excellent wing play to set up De Jong, and went close with two further efforts from distance before departing to a standing ovation.
Cillessen's rotten return
In his first game at Camp Nou since leaving for Valencia in June, Cillessen made only one save in the first hour - and it left Pique with an easy finish on the rebound. It was not a happy homecoming.
Barca begin Champions League duties away to Borussia Dortmund on Tuesday, with a league trip to Granada to follow four days later. Valencia head to Chelsea before a home league meeting with Leganes.