Valencia did manage to halve the deficit three minutes from time, Ousmane Dembele carelessly bringing down Jose Gaya for a penalty which Dani Parejo slotted beneath Marc-Andre ter Stegen.
Ernesto Valverde's side is now 14 points clear at the top of the table, before second-placed Atletico Madrid plays its game in hand against Levante on Sunday.
"These three points are gold", Valverde said.
"The week has been hard and that hard week affects everyone, including the fans".
"This wasn't a week to think about records, we don't give it too much importance. You want to repay them".
Barca have reached the Copa del Rey final, where they face Sevilla next Saturday in Madrid, a game which could see the Catalans come closer to a domestic double.
He could still win a league and cup double this year with Barcelona.
This was already a club record run, now eight games better than the 31 unbeaten Pep Guardiola's brilliant team strung together in the season of 2010-11.
IPL 2018, MI Vs DD: 3 Masterstrokes from the game
Less than 48 hours later, they will be playing at Eden Gardens in Kolkata as the matches keep coming thick and fast in the IPL . In his next over, Tewatia foxed Lewis by floating up a googly and drawing the mis-hit that was caught at mid off.
Up next for Barcelona is a trip to the Estadio de Balaidos where they will face Celta Vigo in a league game on Tuesday.
"Records are there to be broken", Suarez told beIN Sports.
Despite their exertions in Rome, Valverde resisted the temptation to rest key players, with Philippe Coutinho and Paulinho coming in for Ivan Rakitic and Nelson Semedo.
The ex-Liverpool man started the match in an inside right midfield position, and found himself floating into unsafe pockets in the 15th minute before sliding a perfectly timed through ball into Luis Suarez for the Uruguayan to tuck the ball in at the near post. Suarez took the finish first time and swept the ball inside the near post.
A awful back-pass from Barca nearly went pear-shaped as it freed in Rodrigo, who took a touch before forcing Ter Stegen pushing his shot to the crossbar for a corner.
A slip by Umtiti presented Valencia with a clear-cut opportunity to equalise five minutes into the second half.
Samuel Umtiti, who's been in contract negotiations with the club, helped his own cause with this header on a corner kick, also assisted by Coutinho.
Ter Stegen, who is usually so assured with the ball at his feet, was under no immediate pressure before playing a poor pass out which eventually fell to Rodrigo, who failed to capitalize on the massive chance as the German goalkeeper redeemed himself with a fine save tipped onto the woodwork. But Barca's morale-boosting win was never really in doubt.