Betis, who remain 13th following their 13th league defeat of the season, had one back in the 79th minute through Ruben Castro, but Griezmann made sure of the points with his second two minutes later, before Thomas Partey added a fifth in second-half stoppage time.
As a result, there was a first league appearance for Atletico B defender Nacho Monsalve, with the 21-year-old joining Lucas Hernandez at centre-back. Saul Niguez and Fernando Torres, meanwhile, both started in the final third.
As for Betis, Juan Manuel Vargas and Cristiano Piccini were both still injured, which left Martin Montoya and Francisco Molinero in the two full-back positions. Dani Ceballos also kept his spot in the team, while on-loan Chelsea midfielder Charly Musonda played a supporting role to leading scorer Ruben Castro.
The first half-chance of the match came to Atletico in the eighth minute when Augusto Fernandez cleverly picked out Saul inside the Betis box, but the Spaniard's header was always over Antonio Adan's crossbar.
Two minutes later, Adan was forced to make a smart save to deny Griezmann from inside the box and the rebound fell to Koke, who headed onto the top of the woodwork as Betis survived two dangerous moments.
Ceballos had a sighter for Betis in the 25th minute after finding space outside the box, but his effort was always straight at Jan Oblak as an open contest in the Spanish capital remained all square.
Atletico had the lead in the 37th minute, however, when Koke threaded a pass through to Fernando Torres, who lifted his effort over the onrushing Adan with supreme confidence.
The home side then made it 2-0 in the 42nd minute when Griezmann's flick, which was meant for Koke, evaded Adan and simply rolled into the back of the net, much to the embarrassment of the Betis stopper.
Betis impressed for long spells of the first period, but only smart goalkeeping from Adan prevented Torres and Griezmann from adding to the scoring in the final moments of the first 45 minutes at the Calderon.
Betis had a wonderful chance to register early in the second half when Heiko Westermann found space inside the box, but the centre-back's header brought a fine save from Oblak, who tipped the ball onto the post.
Fernando Torres then set up Griezmann in the 55th minute as Atletico came close to putting the game to bed, but Adan was again quick off his line to prevent the Frenchman from adding a third for the home team.
Atletico did have a third in the 65th minute, however, when Angel Correa, who replaced Fernando Torres just past the hour, delivered a cross into Juanfran and the right-back edged the ball home from close range.
Thomas came close to a fourth goal 15 minutes from time when he broke free inside the Betis box, but the midfielder's low strike hit the post as Betis continued to struggle.
The visitors did register in the 79th minute when Ruben Castro made use of a mistake from Oblak before firing home for his 16th league goal of the season, but two minutes later, Atletico re-established their three-goal lead when Correa set up Griezmann, before Thomas made it five with a close-range finish late on.