Diego Simeone's side, third, have won nine matches straight across all competitions and the win pulled them a point behind second-placed title holders Real Madrid and three behind leaders Barcelona.
Atletico showed character and quality as they fought back from two goals down in the second half to triumph.
Rodrigo De Paul powered Atletico ahead but Dodi Lukebakio equalised for 13th-placed Sevilla.
Isaac Romero sent the visitors ahead after half-an-hour and Juanlu Sanchez gave them a 3-1 lead before the hour mark.
However, Griezmann pegged one back before substitute Samuel Lino scored a stunning effort from the edge of the box and the Frenchman then blasted home in stoppage time to continue Atletico's superb run of form.
"We're a united group, all going the same way, and we see that when new players come on in the 60th minute, the 70th, they change the game and that was the case today," Griezmann told Movistar.
"So (it shows) that this is the way to keep dreaming big...
"We will try to win all the games that are left for us, we know there will be difficult moments, but united like this, we can keep fighting."
The otherwise superb Lukebakio gave the ball away for De Paul's opener as he tried to dribble out of a dangerous area.
The Argentine midfielder lashed the ball beyond the reach of Sevilla goalkeeper Alvaro Fernandez from the edge of the box in the 10th minute.
However the Andalusians were quickly on level terms, with Lukebakio making amends two minutes later.
The Belgian forward, on the left of Atletico's area, took one step to beat his marker and smashed home at the near post, catching out Jan Oblak with his powerful strike.
Sevilla played with huge intensity, pressing Atletico and throwing themselves into tackles, up for the fight at the Metropolitano stadium in front of over 60,000 fans.
They took the lead in the 32nd minute when Romero broke in behind to reach Kike Salas' pass to net his first goal of the season.
The striker fired past Oblak, who got a hand to his effort but could not keep the ball out.
The hosts battled hard to find a way back in but Francisco Garcia Pimienta's Sevilla soaked up the pressure and netted their third early in the second half.
They put together an excellent team move, involving Lukebakio, Djibril Sow and Salas, the latter of whom crossed for Sanchez to slam home at the far post.
'Correct those details'
Atletico were unperturbed and swiftly pulled one back through Griezmann, after an excellent through-ball from Spanish midfielder Pablo Barrios.
Sevilla goalkeeper Fernandez made several good saves as the Rojiblancos pushed for an equaliser, denying substitute Alexander Sorloth twice, and the Norwegian striker narrowly missed a tempting cross with a diving header.
Atletico pulled level thanks to Lino's spectacular strike from distance, which was reward for their heavy pressure in the second half.
Sevilla were under the cosh in the final stages and eventually caved when Griezmann swivelled and smashed the winner past the helpless Fernandez.
"We're on a good run (but) we had not been conceding goals and today we let them in easily," said Atletico midfielder Marcos Llorente.
"Today we are happy, but we have to correct those details."
Earlier, Athletic Bilbao tightened their hold on fourth place with a 2-0 victory over Villarreal, fifth.
Aitor Paredes and Inaki Williams struck to help the Basques move six points clear of their opponents, who have played two fewer games.
Bilbao have won four consecutive games in the top flight, including beating champions Real Madrid on Wednesday, and are aiming for Champions League qualification next season.
"We're establishing ourselves in fourth, which is what we wanted," said Williams.
"I think we competed really well, when you're in form, you have to surf the wave."
Villarreal have now gone four matches without victory in all competitions, including a shock Copa del Rey defeat by fourth-tier Pontevedra in midweek.
