Mikel Rico had put the Basques ahead in the 17th minute at a highly charged San Mames, but Griezmann levelled for Atletico right at the start of the second half.
Tiago Mendes then won a hotly-contested penalty which Raul Garcia converted, and with the hosts chasing the game, Griezmann scored twice more on the counter-attack.
Luciano Vietto scored twice as Villarreal claimed a 3-0 win over struggling Deportivo La Coruna to keep the top four in their sights.
Jonathan dos Santos opened the scoring for the Yellow Submarine in the 10th minute before a second-half double from Vietto sealed the win.
Villarreal finished the game with 10 men after Tomas Pina was sent off in the 80th minute for a professional foul on Depor midfielder Alex Bergantinos, but it did not affect the result.
Liverpool loanee Luis Alberto scored a magnificent winner within seconds of coming on as a substitute as Malaga claimed a 2-1 comeback victory at struggling Elche.
The hosts had taken the lead in first-half injury time thanks to a disputed headed goal by David Lomban that looked to be offside, but Malaga hit back just after the break through Ignacio Camacho, and then Luis Alberto hit the winner with 13 minutes left.
In the day’s other game, struggling Granada were held to a 1-1 Primera Division draw by Getafe at Los Carmenes.
Jhon Cordoba handed the hosts the lead five minutes before the break but Emiliano Velazquez struck back deep into the second half to earn Cosmin Contra’s men a share of the spoils.