The visitors began the game in positive fashion and passed the ball around nicely but struggled to create a clear-cut opportunity once inside the box.
In response, the Bhoys managed the first shot on target in the game after 20 minutes when Camilo Duran’s 12-yard header forced a routine save from Ionut Radu.
This chance appeared to give the hosts a boost and their attacking play suddenly had a bit more bite to it, but this fizzled out after the half-hour mark.
What followed was a disappointing spectacle for the last quarter of an hour of the first half, in which both teams cancelled each other out in a disjointed encounter.
Celta would have been encouraged, though, that Ferran Jutgla was able to test Sam Johnstone in goal just before the whistle was blown for the interval.
Martin O’Neill’s side displayed a real sense of urgency after half-time, and they were quickly rewarded for this. The ball fell kindly towards Mika Baur in a very promising position inside the box, and his right-footed effort from 15 yards found the back of the net to give Celtic the lead and secure his second Europa League goal of the season.
With momentum behind them, Benjamin Nygren came close to doubling the Bhoys’ lead 10 minutes later, but his shot from an acute angle was superbly kept out by Radu at his near post.
The Bhoys continued to push hard for a second goal, and that was almost achieved when Liam Scales’ looping header from inside the area bounced off the bar.
Os Celestes desperately needed someone to kickstart their European campaign, and it was Javi Rodriguez who answered the call.
Firstly, after 68 minutes, an excellent delivery into the box from Hugo Alvarez was met by the head of Rodriguez and his effort levelled the contest.
He wasn’t finished there though, as just three minutes later, his superb strike from just over 20 yards found the bottom corner to edge Celta in front for the first time in the game and deflate the Celtic Park atmosphere.
Rodriguez’s second goal really knocked the stuffing out of the hosts, and the Scottish champions could not recover in time to earn a last-gasp draw.
Celtic will be desperate to revive their European campaign in two weeks' time when they host Besiktas, with the Bhoys currently languishing in the bottom three of the league phase without a point to their name.
Meanwhile, Celta Vigo will aim to record back-to-back Europa League victories to boost their hopes of qualifying for the knockout phase when they take on Royale Union Saint-Gilloise at home.
Flashscore Man of the Match: Javi Rodriguez (Celta Vigo)