Costa picked up a controversial yellow card for diving in last week's derby draw with Real Madrid and, although the club attempted to have the decision overturned, the referee’s initial decision was upheld and Villa was left to supply the goods in his absence.
The visitors struggled in the first half as Celta looked solid at the back and relatively threatening in attack, but the title chasers took over and began to dominate after the break.
Barcelona's defeat to Real Valladolid earlier in the day gave Atleti the chance to go top for at least one day, with Real playing Levante at home on Sunday.
Soon after, Villa forced Yoel into a smart 25th-minute save, while Celta were unlucky not to get a penalty on the half-hour mark as Hugo Mallo's powerful strike hit Toby Alderweireld on the arm.
Although both sides appeared to be conservative at the start of the second period, Atleti did worry Yoel in the 51st minute as Garcia's 25-yard effort looked to have been deflected past the goalkeeper, but he reacted quickly to make the save.
Villa latched on to Jonny Castro's disastrous backpass attempt, raced through on goal and confidently stroked past the helpless Yoel.
The lead was doubled just two minutes later with Villa once again to thank as he latched onto Jose Sosa's well-placed cut-back and produced a composed finish.
