France worldwide Griezmann, 27, has now scored on each of his last three league visits to the Bernabeu and has 18 goals in his last 20 games in all competitions.
Griezmann's goal came on his 200th appearance for Atleti while Ronaldo's was the 500th in the 161 league matches played between these two city rivals.
But they broke through early in the second, Ronaldo smashing home in the 53rd minute after Bale had got away down the left.
Real had the better chances and Marco Asensio and Marcelo hit the woodwork at 0-0.
Atletico Madrid were behind for less than five minutes before Griezmann levelled.
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Real were convinced they should have had a penalty after 11 minutes when Toni Kroos went down inside the penalty area following a collision with Juanfran.
Navas stopped Koke's well-taken volley from handing Diego Simeone's side the lead as they responded well to scoring, ahead of a substitute for Real that saw Ronaldo replaced by Karim Benzema.
A game that looked in danger of drifting towards a stalemate sparked into life after half-time with two goals in four minutes. He has a lot of space around him and fired a terrific shot from the edge of the box towards the right side of the goal, but the goalkeeper made a brilliant save and stops this one but the rebound fell to Antoine Griezmann and he fired home from close range to make the scoreline 1 - 1.
Real were more aggressive in the search for a late victor but Atleti dug in.
Saul bent a shot narrowly wide for Atletico and Real, who brought on Luka Modric and Isco for Mateo Kovacic and Asensio, nearly snatched a late, late victor but Oblak made an excellent save from Sergio Ramos' free kick.
Cristiano Ronaldo, fresh from his stunning Champions League brace midweek, took part in the session at their Valdebebas sport facilities along with his team-mates in the Spanish capital.