Griezmann doubled his side's lead with a penalty on the stroke of half-time, but the hosts fired back in the second half to pull level.
The French striker settled the game with 10 minutes left, netting goal number 125 on his 300th La Liga appearance.
The 27-year-old is now on 10 goals in 23 games in all competitions this season, but he feels his pace is dropping as a long 2018, which included a Europa League title and France's 2018 World Cup victory, is taking its toll on him.
"I can still improve, I am not at my best level, it has been a long year and sometimes I lack speed," he told reporters.
"I'm very happy to be able to help the team. It's a very long year, we have a game and then a little rest, there are games when it is difficult to be loose and fast, today was a day like that but I was able to score two goals."
Griezmann was disappointed to see his side come under pressure in the second half after they controlled the opening 45 minutes, especially as coach Diego Simeone told them to take caution.
"We let them focus and we suffered, Simeone had warned us about that at the break," he added.
"It was important to win away from home, which is what he is telling us. In the second half it cost us and the coach made changes that helped us to be better and we went to score a third."
Griezmann's penalty was held up because the referee paused to review a video replay, and the World Cup winner was asked for his thoughts on VAR.
"It's long, but it was the right thing to do, it can make people angry. It's a good thing especially when it's going well for you. When it goes wrong, you get angry."