The midfielder scored the only goal of the game just before halftime as the Liga leader prevailed despite seeing Juanfran sent off late on.
Diego Simeone's side remains at the top of the table after another hard-fought victory, with Garcia's 18th strike of the season in all competitions proving to be the difference.
Much of Atletico's success has been down to their resolute defense – the club has conceded a league-low 22 goals in 35 games – and again proved too tough for an opponent to break down.
The club has now conceded just one goal in its last nine league matches, with the result ending Valencia's scoring run in 25 successive La Liga matches at the Mestalla.
Garcia's 43rd-minute goal – a smart header from Gabi's cross – should have been added to by Diego Costa when he was sent clean through.
The forward was denied and Valencia rallied late, with Jonas seeing a header blocked, but Atletico held on for all three points despite Juanfran receiving a straight red card in second-half stoppage time for a lunge at Pablo Piatti.
The visiting side – which travels to Chelsea for the second leg of their Champions League semifinal on Wednesday – now needs just six more points from the remaining three matches to win the title.
The importance of breaking Atletico's La Liga drought was underlined by Simeone, who picked nine of the players who started in Tuesday's goalless Champions League draw against Chelsea.
Valencia started well, with Paco Alcacer firing wide after a smart turn, while at the other end, the recalled David Villa almost provided a spectacular opener with an instinctive curling shot that just missed the target.
Filipe Luis had a fierce strike deflected wide and Atletico started to exert its dominance, with Koke volleying into the ground and over Vicente Guaita's goal after Costa evaded two players to create the chance.
And the team struck just before halftime, when Garcia beat two defenders and Guaita to Gabi's superb cross, heading into the right corner for his 10th league goal of the season.
Atletico should have doubled the lead just after the break when the prolific Costa was played in by Koke, but he shot straight at Guaita.
A far-post header from Jonas looked destined to find the bottom corner, but Diego Godin was in the way to thwart him, and Sofiane Feghouli saw a blistering effort flash just wide.
Costa had another great chance to score but was denied by Guaita after trying to round the goalkeeper, and Atletico's desperation to maintain the lead saw Juanfran scythe down Piatti in the dying stages.
It did not impact the result, though, as the Rojiblancos won a league match at Valencia for the first time since 2003.