The victory at a festive Calderon, Atletico's 11th in 12 matches this season, trimmed the gap to unbeaten leaders Barca back to one point and restored their five-point advantage over city rivals Real in third.
Champions Barca, 1-0 winners over Espanyol in Friday's Catalan derby, have 34 points, with Atletico on 33 and Real, who beat Rayo Vallecano 3-2, on 28.
Villa, who joined Atletico from Barca in the close season, opened the scoring in the 33rd minute with a well-taken volley and Costa raced on to a Koke through ball seven minutes later and finished clinically past Bilbao goalkeeper Gorka Iraizoz.
It was the in-form Brazil-born forward's 13th La Liga goal of the campaign, putting him level at the top of the scoring chart with Real's Portugal international Cristiano Ronaldo.
Bilbao, who did not manage a single shot on target, played the final 10 minutes with 10 men after substitute defender Erik Moran picked up two bookings and the Basque club stayed fifth on 20 points.
Valencia were the last team apart from Barca or Real to win La Liga back in 2004. Only Villarreal have broken the stranglehold of the world's richest clubs by income on the top two places since, coming second behind Real and ahead of third-placed Barca in 2008.
Atletico have less than a quarter of the annual budget of Real and Barca and a limited squad but have been transformed into genuine title contenders by Argentine coach Diego Simeone since he took over at the end of 2011.
"We had a good rhythm to our play and we managed to dominate the match," Simeone, who was an Atletico player when they last won the title in 1996, told a news conference.
"It fills us with pride to be able to compete, above all for the fans," added the 43-year-old.
Only three teams have recorded 11 wins in their opening 12 games of a La Liga season, Real in 1961 and 1991, Barca in 2012 and 2013 and Atletico in 2013.
The club nicknamed 'the mattress makers' are also flying in Europe and have won all three Champions League Group G matches ahead of Wednesday's home game against Austria Vienna.
Valencia, who are a shadow of the 2004 champions after years of financial struggle, ended a run of three successive defeats and eased the pressure on Serbian coach Miroslav Djukic when they won 1-0 at Getafe in an ugly encounter earlier on Sunday.
Dorlan Pabon struck the only goal in the 41st minute at the Coliseum and the game ended with brawling Valencia goalkeeper Vicente Guaita and Getafe defender Alexis Ruano having to be separated by officials and team mates.
The victory lifted the visitors to ninth with 16 points from 12 matches while Getafe stayed sixth on 19.
"We were in a tough situation but the team knew how to respond at a difficult stadium against a team that is playing very well," Valencia midfielder Dani Parejo told Canal Plus television.
"I think we are coming away from here in much better shape. The coach told us to stay calm, ignore the pressure and just enjoy the game."
Malaga left it late to secure a 3-2 victory at home to Real Betis when substitute Samuel Garcia Sanchez volleyed in from close range in the 94th minute.
The match at the Rosaleda was delayed for several minutes early in the second half when Betis defender Damien Perquis appeared to be knocked out by a clash of heads with Malaga forward Fabrice Olinga.
An ambulance was brought on to the pitch to take Perquis to hospital and initial reports in local media suggested he had regained consciousness and was undergoing tests.
Malaga are 12th on 13 points, with Betis mired down in 18th on nine and coach Pepe Mel's situation looking increasingly precarious.
The start of the second half of Granada's game at Levante was set back by around 10 minutes by a lighting failure before the visitors snatched a 1-0 victory when goalkeeper Keylor Navas could only palm Piti's 91st-minute free kick behind him into the net.
Levante are eighth on 17 points, while Granada climbed to 11th on 14.
Villarreal, who have 20 points in fourth, can move three points clear of Bilbao with a win at promoted Elche on Monday.