Dorlan Pabon's rocket late in the first half was enough to give under-fire coach Miroslav Djukic some breathing space.
Getafe matched their opponents for most of the game and had several chances to score, the best of which fell to Jaime Gavilan in the last minute.
The win moved Valencia up to ninth in the Primera Division, three points behind Getafe.
Valencia had lost at home to bottom-side Almeria in midweek and the tension at the club was so high that the players and coach were told not to speak to the media before the trip to Getafe.
The Madrid side, meanwhile, had beaten fourth-placed Villarreal to move up to sixth in the table.
And Getafe were more adventurous early on.
First Pedro Leon tested Diego Alves from 25 yards out, but the goalkeeper was able to get down and make the save.
His next stop was far more spectacular. Angel Lafita found space down the left wing and crossed for Ciprian Marica to head at goal, but Alves was able to tip the ball up into the air before catching it.
That was the keeper's last contribution, though, as he was forced to come off with an injury after 13 minutes, replaced by Vicente Guaita.
Valencia then started to get in the game.
First Dani Parejo forced a save from Miguel Angel Moya from a free-kick, then Paco Alcacer fired a shot against the bar as he tried to convert a cross from Barragan inside the area.
Alcacer had another chance to score on the half-hour mark, sliding in but failing to meet a deep, low cross from Pabon.
Despite their advances, it took a truly special goal to break the deadlock.
Pabon received a short pass 35 yards from goal and rifled a shot straight into the top corner, despite slipping as he took the shot.
The second half saw more chances than the first and as the game progressed Getafe came closer to an equaliser.
Leon had their first clear chance of the second period, arriving in the left hand side of the area to shoot at goal, but Guaita was able to stop the shot with his legs.
Pablo Sarabia also caused Guaita problems with a stinging long-range effort, but the ball was cleared after the keeper punched it clear.
Luis Garcia's side poured men forward in the final minute and were nearly punished at the end.
Valencia broke quickly and Federico Cartabia raced clean through with just Moya to beat and Alcacer in support to his left, but the keeper grabbed the ball from under his feet before he could get the shot away.
But in the final minute of injury time Getafe squandered a golden chance to equalise.
Substitute Diego Castro got down the right and pulled the ball back for Jaime Gavilan, who shot woefully off target.