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.
Villa netted his first in fine fashion in the 62nd minute as he pounced on a dreadful backpass before coolly slotting into the bottom-right corner, and he doubled his tally a couple a couple of minutes later with a tidy finish.
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.
Following successive away defeats in the league, Atleti coach Diego Simeone conceded his side cannot afford to lose again if they are to continue their title challenge this season and they produced a professional display in Vigo, though they did have things all their own way early on.
Celta enjoyed an encouraging start as Nolito forced Thibaut Courtois to rush off his line and make a last-ditch challenge in the fifth minute, before captain Borja Oubina dragged wide from the edge of the area a few moments later.
Alex Lopez was the next to go close for Celta as he fired just off target after cutting in from the left flank, but Atleti did finally wake up as Raul Garcia hit a half-volley on to the roof of the goal from 40 yards out on the right wing.
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.
As the hour mark approached, Atleti began to look dominant and they deservedly took the lead 62 minutes in.
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.
The quickfire double killed off the spirited visitors who remain in mid-table after a first home defeat in eight outings.
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