A floodlight failure at the start of the second half delayed proceedings by about 20 minutes but it did not dim Celta's attack, who forged a number of scoring opportunities which were each wasted in turn.
Papakouli Diop then made Celta pay for their lack of a clinical finish when he snatched an 89th-minute winner to claim what had seemed an unlikely three points for the visitors.
Celta's first chance came through Santiago Mina in the 16th minute when he forced a brilliant save from Levante goalkeeper Keylor Navas before Nolito curled a shot just wide of the post.
Michael Krohn-Delhi then put a powerful effort wide as Celta could have arguably been 3-0 up in the opening half an hour.
Toni tested Navas again shortly before the break but the Levante stopper ensured it remained goalless at the break.
The match was then thrown into doubt at the start of the second half as a power cut threatened to cause its postponement, however the game resumed, and Celta picked up where they had left off.
Mina fired a shot high and wide before Charles lacked direction with his strike and substitute Fabian Orellana wasted another clear-cut chance as he also sent the ball wide.
It was then that Levante struck, totally against the run of play, when Diop volleyed home from 25 yards.
Levante had four minutes of added time to see out but they did so despite Rafinha's best efforts with a testing cross to claim an unlikely win.