Ezequiel Garay put Valencia into an early lead and although Lionel Messi quickly equalised for the Catalan giants, they did not do enough in the final third to break down a resolute Valencia defence for a second time.
Barcelona drop down to second place after Sevilla's win over Celta Vigo earlier in the day, while Valencia are now 14th after a sixth draw in eight league matches.
Valencia went into this match having struggled for goals in recent weeks, but it took the home side less than two minutes to go in front against last season's champions.
An inswinging corner was not dealt with by Gerard Pique, who only glanced the ball into the path of Garay who converted into an empty net from two yards at the back post.
Michy Batshuayi then had a low shot tipped behind by Marc-Andre ter Stegen and Geoffrey Kondogbia also went close with an effort from long range, but Valencia were soon dealt a huge blow when Goncalo Guedes was forced off with a knee injury.
That prevented the home side from building on their early momentum and Barca gradually eased their way into the game before finding an equaliser midway through the half as Messi exchanged a one-two with Suarez before finding the bottom corner from 20 yards out.
The visitors had the better of the game heading into the break but Ernesto Valverde's outfit were struggling to create much space in the final third, and Valencia almost regained the lead on the break with Jose Gaya lashing a well-struck effort marginally wide of the far post.
Valencia went onto the front foot after the restart with Denis Cheryshev sending a shot just over with the outside of his boot, while Philippe Coutinho went close with a long-range strike at the other end.
Messi almost capped a fine team move with a second goal, only for Neto to keep his shot out at the near post, but that proved to be Barca's last opening until the final quarter of the game as Valencia showed more ambition going forward.
Barcelona continued to dominate possession but they were struggling to find a way past the impressive Garay and Gabriel Paulista, who were leaving Neto will little to do between the sticks.
Neither team were able to create anything during the final 10 minutes, and the game quickly petered out with both clubs setting for a point apiece ahead of the international break.
VALENCIA (4-4-2): Neto; Piccini, Garay, Paulista, Gaya; Soler, Parejo, Kondogbia (Coquelin 79'), Guedes (Cheryshev 12'); Gameiro (Rodrigo 67'), Batshuayi
BARCELONA (4-3-3): Ter Stegen; Semedo, Pique, Vermaelen, Alba; Rakitic, Busquets, Arthur (Rafinha 88'); Coutinho (Dembele 84'), Messi, L.Suarezbody check tags ::