There was some shaky defending from Barca in the opening minutes of the second half as Samuel Umtiti lost his footing to allow Santi Mina to pick up the ball and feed Rodrigo.
Guedes forced Marc-Andre ter Stegen to claw away a fine strike during a sluggish opening spell from Barca, but Suarez soon had the fans on their feet.
The Catalan side are still on track to win a domestic double, though, as they sit top of La Liga and face Sevilla in the Copa del Rey final next weekend.
Barca rode their luck at times and had goalkeeper Ter Stegen to thank for keeping an impressive Valencia at bay until the closing minutes of the game.
Coutinho was afforded too much time and space just outside the penalty area, allowing him to roll an inch-perfect pass into the path of Suarez who calmly slotted past Murara Neto at his near post.
Iniesta then saw his 71st-minute scissor-kick well kept out by the visiting stopper ahead of a spell of pressure from the away side. "After a defeat like the one against Roma, there's always a painful period when questions are asked and everyone is affected".
As a result, Barça have surpassed Real Sociedad's respective record of 38, which they had set in the 1979-80 season in La Liga. Barcelona created their first chance in the eighth minute but, Luis Suarez mistimed his effort, 12 yards out.
It's not insane to think Burnley could climb into sixth and get an automatic space in the UEL, either.
It's both an impressive and an entirely unexpected record considering the nightmare summer that the Catalans suffered with the departure of Neymar.
"We played a good match today, too bad we were missing our finishing touch", Valencia coach Marcelino Garcia Toral said.
"When Barcelona play at home, the numbers say that it is the most difficult venue in the league", he explained. But Barca's morale-boosting win was never really in doubt. Then Barcelona took a 2-0 lead and suddenly Andrés Iniesta came to life. Maybe he should anyway, and full marks to Burnley for holding onto their man - and Dyche staying put - despite relegation.
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