Winning without playing in top gear is often heralded as a hallmark of champions, and Diego Simeone's side are well-placed for the title on that basis, with Real Madrid's 2-1 defeat at Sevilla putting them clear at the top.
Atleti struggled to break down their opponents in a frustrating first half that saw Arda Turan go closest.
Real's defeat means Atleti are now a point clear of Barcelona as the advantage in the title race continues to swing between the three teams.
But it is the season's surprise package Atleti who now appear in control after a 17th-consecutive home league game without defeat.
Turan had the hosts' best chances in the opening stages, first prodding wide after David Villa's near-post flick was blocked and then skewing a poor effort off target from 20 yards.
The atmosphere dropped as Thibaut Courtois was forced to keep out Yacine Brahimi's fierce effort and the visitors were quick out the blocks in the second half, Recio and Jeison Murillo somehow failing to turn Fran Rico's corner home.
Having seen Roberto keep out a firm header from Gabi's corner just two minutes earlier, Costa met another delivery from the home captain and got his effort beyond the veteran goalkeeper to raise the volume in the Calderon once again.
Roberto did well to deny Gabi and substitute Diego as Atleti pushed for a second, but Atleti were ultimately left to settle for the narrow scoreline.
Defeat for Granada keeps them looking over their shoulders, just six points clear of the bottom three, while the hard-fought three points could prove crucial come the season's conclusion for Atleti as they seek their first league crown since 1996.
