Atletico remain second on 68 points, four ahead of Madrid who occupy the third spot with seven games left to play in the league.
Madrid had more chances, although Atletico also threatened on a number of occasions and the lively Griezmann became the first player this century no score in three consecutive Liga games for a visiting side at the Bernabeu.
In comparison to recent versions of this fixture, which have included Champions League finals, Cope del Rey finals and league games with both sides vying for the championship, this at times had a rather low-key feel.
The 33-year-old was withdrawn after 63 minutes for a rest ahead of Madrid's Champions League quarter-final second leg against Juventus at the Bernabeu on Wednesday and was an animated observer from the bench as Sergio Ramos twice came close to a late victor - once with a free-kick pushed over by Jan Oblak and then with a header from the resulting corner.
Real had several openings around the penalty area but a combination of poor finishing and strong blocks, from Oblak in particular, kept them at bay. However, the referee waved away the German playmaker's protests.
And Diego Simeone's side nearly netted immediately after, but Koke's attempt was brilliantly kept out by the Costa Rican global.
A game that looked in danger of drifting towards a stalemate sparked into life after half-time with two goals in four minutes.
Bale had operated on the margins of the contest up until that point but the Wales winger stormed down the left flank and Ronaldo pulled into space behind Lucas Hernandez, drilling a volley into the turf and beyond Oblak.
But Atleti were soon level.
Thomas Partey played in Griezmann, who cleverly slid possession to Vitolo and was on hand to steer the loose ball into a vacant goal after Navas saved at the former Sevilla winger's feet.
Surprisingly, Ronaldo was withdrawn soon after that goal, much to the bemusement of the Bernabeu faithful, and with the Portuguese off the pitch, free-kick duties were left to Ramos who nearly snatched all three points in injury time when his set-piece destined for the top corner was clawed out by Oblak.
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.