The lead only lasted five minutes though as Messi released Cristian Tello to curl home a fine equaliser and the roles were reversed 18 minutes from time when Tello cut the ball back for Messi to drill low into the net.
The goal was Messi's 43rd league goal of the campaign and also secured another record for the World Player of the Year.
By scoring for the 19th straight game, he became the first player to score against every other side in the Spanish league in consecutive matches.
However, Celta gave their chances of survival a huge boost as Oubina headed home Fabian Orellana's cross at the back post two minutes from time.
And Barca assistant manager Jordi Roura admitted that his side lacked the concentration required to see the game out.
"I don't want to point out individual errors," said Roura, whose team are still 13 points ahead at the top of La Liga after Real Madrid were held 1-1 by Real Zaragoza.
Real held Second-placed Real Madrid failed to make up ground on Barcelona as they could only manage a 1-1 draw with Real Zaragoza.
Barca's draw with Celta Vigo had handed Madrid the chance to cut the Catalans' lead at the top to 11 points, but Zaragoza were more than worthy of their point as they moved three points clear of the relegation zone.
Rodri put the hosts in front after just five minutes with a calm finish, but Madrid were level before the break as Cristiano Ronaldo equalised with a typically powerful effort from Gonzalo Higuain's pass.