Angel Lafita gave Luis Garcia's side the lead after 20 minutes but Getafe failed to cling onto their advantage and made things easier for Celta when Lisandro Lopez was shown a second yellow card early in the second half, shortly after which Rafinha equalised.
The point means Getafe are still four points from the relegation zone while Celta remain in 10th.
The two teams went into the match in contrasting form, with Celta picking up 10 points from their previous four league matches, while Getafe were without a win since the end of November.
But it was Garcia's side who made the stronger start.
With just two minutes on the clock Pedro Leon connected with a Pablo Sarabia pass on the counter-attack, forcing a save from Celta goalkeeper Yoel Rodriguez.
The former Real Madrid midfielder got in shooting range again a few minutes later, but this time blazed over the crossbar.
The visitors' pressure eventually led to the opening goal, Juan Rodriguez punting a free-kick up field to Lafita, who brought the ball down with one touch before firing into the bottom corner.
Celta had only created half chances up to that point but responded well to going behind, peppering their opponents' goal until Rafinha tested Getafe stopper Miguel Angel Moya, Fabian Orellana struck the crossbar, Augusto Fernandez shaved the post and then a mis-hit effort from Orellana had Moya scrambling across his line to prevent the ball going over.
Getafe were fortunate to make it into their dressing room at half-time still ahead but a Celta equaliser looked inevitable once Lisandro was sent off for a second bookable offence on 53 minutes, giving the hosts a one-man advantage for the remainder of the game.
Garcia reacted quickly by bringing on defender Rafa in place of Sarabia, but the change was unable to prevent Rafinha levelling just six minutes later.
The Barcelona loanee ran with the ball to the edge of the area before unleashing a shot that flew into the net.
With the balance of the game in their favour, Luis Enrique's side naturally went on the hunt for a winner.
Augusto Fernandez was denied by Moya when the Argentinian had just the goalkeeper to beat, and Nolito hit the post in injury-time, meaning Celta would have to make do with just the one point.