Pepe opened the scoring for Real with a looping header in the 52nd minute and it looked like the hosts would earn the victory that would have moved them five clear of Barcelona at the top of La Liga.
But Griezmann's equaliser ensured that the gap remained three points with Barcelona having the chance to go back top on goal difference with victory at Malaga later on Saturday, although they have played a game more than Real.
Fernando Torres had missed a one-on-one chance for Atletico as they sought their equaliser but they eventually grabbed the point when substitute Angel Correa put in Griezmann with a sharp pass and the Frenchman finished expertly.
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Despite needing a late leveller, Atletico manager Diego Simeone said his team deserved to take the three points from the derby clash.
"Our changes gave us more agility in the middle of the pitch. We freed Correa and are left with a bitter taste because in the end we were close to winning,” he said.
His Madrid counterpart, Zinedine Zidane, however, put the dropped points down to a lapse in concentration.
“It was a great performance (from Real), but we just lost concentration for five minutes.
“I can’t say that Atletico played well. We didn’t get the second goal and we suffered.”
The first half had been a tense affair, with both teams playing patiently and carving out a couple of opportunities to take the lead.
Atletico goalkeeper Oblak denied Cristiano Ronaldo at his near post and did brilliantly to keep out a fierce Karim Benzema strike.
He was eventually beaten by Ronaldo but a stunning headed clearance off the line from Stefan Savic kept the score level.
Griezmann had a fine chance after Sergio Ramos blundered by passing the ball straight to him, but his strike was tipped wide by Keylor Navas.
Oblak made another terrific save to deny Benzema early in the second half, before Pepe opened the scoring.
The game was petering out but Griezmann's late intervention earned third-placed Atletico a point.
Opposing manager Zidane had words of praise for the Atletico striker, who scored his 23rd goal of the season in all competitions.
“Don’t ask me how to stop him, you just have to admit that he’s a great player who scores goals,” said the Frenchman of his compatriot.
“He has scored a lot of goals this season and he’s got another one today."
(Editing by Toby Davis and Ian Chadband)
