Griezmann used his left foot to drive home a perfectly placed throughball by substitute Angel Correa to equalize at the Santiago Bernabeu in the 85th minute.
Pepe put Real ahead in the 52nd, when he shook off his marker and headed Toni Kroos' free kick just inside the far upright.
But Real failed to press for another goal and Griezmann capitalized on its complacency.
"We can't be happy today," Madrid coach Zinedine Zidane said. "We could have easily taken the three points. The league title will go down to the wire."
The stalemate in the Spanish capital derby left leader Real three points ahead of Barcelona, which visited Malaga later Saturday. Real also has a game in hand, but faces matches against Barcelona and fourth-place Sevilla over the final two months.
Atletico, which added the draw to wins at Real in the last three league campaigns, stayed in third place, 10 points back.
Real was on the brink of taking a major step toward its first league title in five years.
That was until Correa, who had gone on in the 61st for Saul Niguez, saw the sliver of space between Real defenders Nacho Fernando and Dani Carvajal. Correa slid the ball forward between the pair, and Griezmann did the rest, slotting his shot past Keylor Navas for his third goal against Real in Atletico's red-and-white.
"We know we are a strong team, with a great mentality. It was tough, but we got the draw thanks to a great pass by Angelito," Griezmann said. He dedicated his 23rd goal in all competitions to his daughter who turned one on Saturday.
While highlighting the effort by his defense, Atletico coach Diego Simeone lauded Griezmann for his "passion, work ethic and ability to move in the box."
But the best praise came from the rival manager.
"You can't wonder how to stop him," Zidane said about Griezmann. "You just have to admit he is a great player."
Atletico can turn with confidence still high back to the Champions League, where it hosts Leicester on Tuesday in the quarterfinals.
Real's six-match win streak ended. It travels to Bayern Munich for its European quarterfinal on Wednesday.
It was a vintage performance by Simeone's Atletico. His team bent but never broke under the steady pressure by the star-studded Real.
Besides Griezmann and Correa's moment of inspiration, Atletico's solid performance was based on the superb saves of Jan Oblak.
The Slovenian goalkeeper turned back two clear chances by Karim Benzema, and another two by Cristiano Ronaldo and Gareth Bale.
And when Ronaldo finally beat him in the first half, defender Stefan Savic hustled back to clear his goal-bound strike with his head.
Zidane's team can barely be blamed for letting up near the end, given that Atletico had only mustered shots by Griezmann and Fernando Torres that Navas had dealt with.
That was when Correa connected with his forward to flip the match on its head and cast doubt on Real's march toward the trophy.
"We played a great game against a great rival, but we dropped our guard for one moment," Real midfielder Casemiro said.
MALAGA 2, BARCELONA 0
BARCELONA, Spain (AP) - Nothing went right for Barcelona.
Neymar dealt a self-inflicted blow to Barcelona's Spanish league title defense when he was sent off with 25 minutes left in its 2-0 loss at Malaga on Saturday.
Lionel Messi sent free kicks well over the bar, Andres Iniesta lost an easy ball in midfield, and Neymar got booked for not giving enough space for a free kick while tying up his laces.
Then there was Malaga's goal, when Jeremy Mathieu tried to catch Sandro Ramirez offside. Instead, the former Barcelona forward was left with half the pitch to bear down on goalie Marc-Andre ter Stegen and fire inside his near post in the 32nd.
And just when Barcelona started to click, Neymar got a second booking for charging into Diego Llorente.
Substitute Jonathan Rodriguez added Malaga's second goal in stoppage time with Barcelona searching for an equalizer.
Barcelona failed to take advantage of the opportunity Atletico Madrid gave it to pull level with leader Real Madrid, where Antoine Griezmann's late equalizer grabbed a 1-1 at the Santiago Bernabeu hours earlier.
Barcelona's setback in Malaga meant Real increased its lead to three points despite the disappointing finish to the Spanish capital derby.
Real, which has a game in hand, still faces matches against Barcelona and fourth-place Sevilla over the final two months as it seeks its first Liga title in five years.
SEVILLA 4, DEPORTIVO LA CORUNA 2
Sevilla got a goal in the first minute by Stevan Jovetic, and overcame a double by Gael Kakuta, to end a six-game winless streak.
Sevilla honored sports director Ramon "Monchi" Rodriguez, who is leaving the club this summer, with a tribute before the match.
ESPANYOL 1, ALAVES 0
Pablo Piatti scored his 10th league goal to give Espanyol a fourth consecutive home victory and keep it in the running for a Europa League berth.