The Catalan giants lost on their last two visits to Celta and the hosts, coached by former Barca goalkeeper Juan Carlos Unzue - who was Luis Enrique's Camp Nou assistant - ensured the visitors' difficulties in Galicia did not let up.
Barca's lack of cohesion was clear throughout, although they took an early lead thanks to Jose Arnaiz, who tucked away an Andre Gomes cross to open the club's account for 2018.
But Celta equalised before half-time, Pione Sisto taking advantage of slack defending to slam home a powerful drive from the rebound after Iago Aspas hit the crossbar and the ball ricocheted off goalkeeper Jasper Cillessen.
Arnaiz spurned chances to put Barca back in front and Valverde turned to Dembele with 20 minutes remaining, but the club-record signing could not inspire his side to victory, although Sergi Roberto turned Nelson Semedo's cross on to the outside of the post in injury time.
Aspas had the first chance inside seven minutes, finding himself through on goal, but Semedo's pace allowed the right-back to make a fine recovery block.
Arnaiz had scored in two consecutive Copa del Rey appearances and he took his chance again in the absence of Luis Suarez and Messi, putting Barca in front in the 15th minute.
The visitors won the ball on the edge of their own area and broke rapidly down the left channel through Denis Suarez, before Gomes pulled the ball back for Arnaiz to drill home a first-time finish.
Celta drew level in the 31st minute, however, Sisto reacting quickest to lash in from a tight angle after a close-range Aspas strike crashed against the bar and Cillessen.
The hosts went close five minutes prior to the break, Daniel Wass driving narrowly wide, before Paulinho thumped a low shot off-target at the other end.
Arnaiz should have scored his second in the 56th minute, sending a diving header from Lucas Digne's cross wide.
Denis Suarez then missed an even better chance at the back post, Semedo's centre begging to be finished but somehow put wide by the midfielder.
Barca were cranking up the pressure and a placed Sergio Busquets shot clipped the bar with Sergio Alvarez well beaten, before Dembele came on for his first appearance in over three months.
Emre Mor drilled inches wide in the 72nd minute, while Alvarez had to save a poked Dembele effort and an ambitious strike from substitute Ivan Rakitic.
Dembele looked sharp on his return, but despite heavy pressure from Barca - Roberto hitting the post in stoppage time - they could not force a winner, leaving the tie in the balance ahead of next Thursday's return leg at Camp Nou, where Celta drew 2-2 last month.