Ernesto Valverde's men looked good value for all three points when Luis Suarez put them 2-1 up just after the hour mark, but Gomez struck soon after to ensure Barca drew a second successive league match following the 1-1 stalemate with Valencia last time out.
Iago Aspas gave Celta the lead, punishing Barca's slow start by rounding off an electric counter-attack with an easy finish, but that only seemed to jolt the home side into life.
Lionel Messi restored parity soon after with a clever finish, before then also hitting the post as Barca took charge.
Barca had to wait to take full advantage of their superiority again, but they did eventually go ahead just after the hour mark, with Suarez rounding off a fine move to become one of the 10 leading scorers in the club's history.
Gomez drew Celta level, though, and ultimately secured the visitors a point despite the home side's late pressure, giving the chasing pack hope of gaining on Barca at the top of the table this weekend.
Although Barca afforded Celta few opportunities to attack in the early stages, the home side lacked urgency in the final third and appeared to be coasting.
They were then carved open by the visitors in the 20th minute.
Aspas was released in behind the Barca defence on the right, charged into the area and squared to Gomez, whose initial effort was brilliantly saved by Marc-Andre ter Stegen, only for the Uruguayan's strike partner to knock in the rebound.
But that goal seemed to be exactly what Barca needed, as they increased their intensity and levelled within two minutes – Messi firing through Ruben Blanco's legs after Suarez stepped over Paulinho's defence-splitting pass.
Barca attacks started to come thick and fast, and Celta had to rely on the post to save them just after the half-hour mark, as Messi charged into the penalty area and shot against the left upright from a tight angle.
The hosts' dominance continued into the second half and Paulinho should have put Barca ahead shortly after the restart, shooting into the side-netting after skipping past Ruben.
Celta did not get so lucky in the 62nd minute, however.
Messi picked out Jordi Alba's run into the left side of the area with a sublime pass, and the full-back squared to Suarez for a straightforward finish.
But their lead did not last long.
Aspas took full advantage of Samuel Umtiti pulling up with an injury and darted up the flank, before dragging a pass back to Gomez, who beat Ter Stegen in confident fashion.
Barca piled men forward in search of a late goal and almost got one through Gerard Pique, but the centre-back hit the post despite dribbling past Ruben.
And then Messi inexplicably sent a free header wide of the mark four minutes from time, as Celta held on for a positive result.
