Messi levelled midway through the first half with a wonderful strike from outside the penalty area, although Barca, who beat Tottenham 4-2 in the UEFA Champions League in midweek, could not find a winner.
Pressure on Ernesto Valverde will grow over the international break, after which Barca will travel to LaLiga's surprise leaders Sevilla, who moved top with a 2-1 win over Celta Vigo.
Valencia led in the second minute, Garay arriving at the back post to tap in after a Dani Parejo corner was inadvertently flicked on by Gerard Pique and Thomas Vermaelen.
Michy Batshuayi and Geoffrey Kondogbia both came close to doubling Valencia's lead as Marcelino's men made a fine start, although Goncalo Guedes' injury curbed their potency.
Barca levelled in the 23rd minute, Messi drilling home from 25 yards after a one-two with Luis Suarez that involved a nutmeg on the helpless Kondogbia.
Kevin Gameiro crashed a shot just wide in the 61st minute as Valencia continued to threaten, then Jose Gaya made a sensational challenge to deny Philippe Coutinho.
Messi fired into the side netting and Suarez saw a brace of late penalty appeals rejected, and problems are mounting for Barca and Valverde after missing the chance to reclaim top spot.