Kevin Gameiro came off the bench to make the breakthrough before Diego Costa netted late on as Atletico recorded their 16th league win of the campaign, which moved them 13 points clear of fourth-placed Real Madrid.
Atletico had the first serious opportunity of the match in the 14th minute, but Jose Gimenez could only turn a clever Griezmann free kick wide of the far post.
Costa had a huge penalty shout in the 19th minute when he went down inside the box under a clumsy challenge from Unai Nunez, but the referee was not interested - much to the frustration of the centre-forward.
Williams was the Bilbao player causing problems in the final third, but the number 11 did not have much to feed on in the opening 30 minutes as Atletico continued to dominate the possession.
Chances were few and far between, however, with the two teams heading down the tunnel at 0-0 following a largely uneventful first 45 minutes of action at the Wanda Metropolitano.
Costa had Atletico's first opportunity of the second period when he met a deep cross from Vrsaljko, but Bilbao goalkeeper Kepa Arrizabalaga was on hand to make a smart save.
Costa had another chance just before the hour when he met a flick from Koke, but the striker could only head over the crossbar from close range.
Atletico boss Diego Simeone introduced Gameiro in the 59th minute in a bid to turn the tide, and moments after the alteration, Angel Correa brought a smart save from Kepa. Griezmann then sent one over the Bilbao crossbar in the 64th minute as the pressure continued.
Finally the breakthrough came in the 67th minute when a counter-attack ended with Griezmann feeding Gameiro, who picked out the bottom corner in impressive fashion to register his eighth goal of the 2017-18 campaign.
Diego Godin headed over the Bilbao crossbar in the 75th minute as the home side looked for a second, and it eventually came in the 80th minute when Costa drove into the visitors' box before finding the bottom corner with a smart finish.
ATLETICO (4-4-2): Oblak; Vrsaljko, Gimenez, Lucas (Godin 46'), Luis; Koke (Gameiro 59'), Saul, Thomas, Correa (Gabi 70'); Griezmann, Costa
ATHLETIC BILBAO (4-3-3): Kepa; Lekue, Nunez, Martinez, Saborit; Vesga (Cordoba 72'), San Jose, Benat (Iturraspe 65'); Sabin (De Marcos 69'), Williams, Susaeta