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.
But top scorer Diego Costa came to their rescue after 63 minutes with his 24th Liga goal of the season.
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.
Simeone had called for a vocal backing from home support ahead of the game, and they did their best to generate an atmosphere, despite there being little to shout about on the pitch.
Youssef El-Arabi nodded a Granada corner wide and Atleti midfielder Tiago saw a goal chalked off for a foul in the build-up as things heated up before the break.
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.
But Atleti finally got their act together just after the hour as Costa put them ahead.
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.
Atleti looked to see the game out with a much more disciplined showing after the goal, and the players may have been buoyed as a roar went around the ground 75 minutes in as news of Real going behind filtered through the crowd.
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.
