Pepe's opener had looked likely to give the home side all three points and a major step towards a first LaLiga title since 2012, only for Griezmann to level matters five minutes from the end of normal time.
Atletico are the only team to have faced Zidane's Madrid away from home and kept a clean sheet, while Diego Simeone's side had won on their last three visits in the league, but they were second-best for the majority of a sun-baked encounter.
It took strong saves from Jan Oblak and a remarkable clearance from Stefan Savic following a Cristiano Ronaldo strike to keep the scores level in the first half, but Pepe made the home side's pressure count with a fine header early in the second.
Atleti had offered little by way of a serious threat to the Madrid goal but they found an equaliser in the closing stages, with Griezmann slotting home his 15th league goal of the season.
The draw sees Madrid move three points clear of champions Barcelona, who face Malaga later on Saturday, but frustrations around the Bernabeu were clear at the end of Zidane's 50th league match in charge.
4 - Real Madrid are winless in their last four league games at home against Atletico (D1 L3), their worst run without winning vs them. Shock pic.twitter.com/80BPq0gTq5— OptaJose (@OptaJose) April 8, 2017
Saul Niguez glanced an early header past Keylor Navas' far post but it was Oblak who made the first save of the match, getting down low to his right to stop a fierce strike from Ronaldo from the edge of the area.
Madrid looked the more settled on the ball without posing too many concerns for the Atleti back four, although Oblak had to stick out a strong arm to keep out Karim Benzema's strike after a clever one-two with Ronaldo.
Oblak was beaten on the half-hour mark but Savic came to Atletico's rescue, the centre-back heading Ronaldo's strike from close range up and over the crossbar from almost on the goal line.
Griezmann had been starved of service but his best chance of the first half came from a poor Sergio Ramos pass, the forward driving towards goal after picking up the loose ball before testing Navas with a low strike from 25 yards out.
Madrid resumed their dominance of the ball straight from kick-off in the second half and could have scored twice in the opening three minutes. Ronaldo nodded a Marcelo cross wide before Benzema was thwarted by a good block from Oblak after he turned his strike partner's header goalwards.
But Oblak could do nothing to stop Madrid taking the lead on 52 minutes through an unlikely source. Toni Kroos swept in a fine free-kick from the right and Pepe powered his header back across goal and into the corner of the net.
Dani Carvajal rifled over moments later but Fernando Torres should have levelled matters just before the hour mark, the veteran striker shooting straight at Navas after racing in behind Ramos to meet Yannick Carrasco's pass.
Pepe gave Zidane some cause for concern 23 minutes from time when he was forced off following a collision with Kroos on the edge of the area, while Simeone threw on Angel Correa in an effort to breathe life into Atleti's attack.
Griezmann forced a good save from Navas with a spectacular overhead kick - although the offside flag had already been raised - but the France star found the net with his next opening, firing low past Navas from 12 yards to rescue a point after Correa had threaded a fine pass between Carvajal and substitute Nacho.
Key Opta stats:
- Pepe has scored his second goal in a derby against Atletico (19 appearances in total), the first one in La Liga (14).
- Real Madrid have scored more goals from set-piece situations than any other side in La Liga this season (25).
- Toni Kroos has provided more assists from set-pieces than any other player in Europe's top five leagues this season (eight).
- Keylor Navas has conceded more goals against Atletico as Real Madrid's keeper than against any other side (eight - all competitions).
- Real Madrid have found the net in 52 consecutive matches in all competitions, the best run ever for a La Liga side.
- Angel Correa has made more assists as a substitute than any other player in La Liga this season (four).