Both teams had chances to make the breakthrough in the derby game but neither could find the back of the net as the points were shared on matchday seven.
Five of the last six La Liga meetings between the two teams have now finished level, and the result keeps Real Madrid top of the table on 15 points - one point clear of third-placed Atletico.
Bale again came close in the 18th minute as he headed over the crossbar from inside the Atletico box, and the Welshman, like Hazard, was finding a lot of space when the ball was turned over in midfield.
Atletico had a golden chance to take the lead on the 30-minute mark when Thomas Partey combined with Kieran Trippier in a wide area before delivering a dangerous low cross towards Costa, but the Brazilian-born Spain international could not make contact at the far post.
Real Madrid's next attempt came in the 37th minute when Toni Kroos found some space outside the box before releasing a strike, but the Germany international's effort was impressively saved by Jan Oblak between the home side's sticks.
Felix then came close down the other end as the summer arrival worked some space outside the Los Blancos box before firing a low attempt just wide of Thibaut Courtois's goal.
The final chance of the first period also came for Atletico when Trippier found all sorts of room inside the away team's penalty box, but the Englishman's low cross was expertly palmed clear by Courtois.
Real Madrid had more possession and more shots in the opening 45 minutes of action, although neither team really did enough to deserve to take a lead into the half-time interval.
Atletico had a big chance to take the lead early in the second period when Costa delivered a dangerous cross onto the head of half-time substitute Angel Correa, but the Argentine could only lift his effort over the crossbar.
Bale then struck high of the target down the other end as the away team continued to have their moments, although it was still an incredibly even contest at the Wanda Metropolitano.
Real Madrid's Welsh attacker had an even better opportunity just before the hour when he raced onto a pass from Nacho, but once again the 30-year-old could not test Oblak.
Saul Niguez flicked a corner just over the Real Madrid crossbar in the 72nd minute - moments after Modric had fired an attempt wide of the post after breaking into the penalty box.
The visitors had another big chance in the 76th minute when Benzema met a brilliant cross from Nacho, but the Frenchman's header was kept out by the diving Oblak.
Koke then turned a Trippier cross over Courtois's goal down the other end with both teams having openings to make the breakthrough in the latter stages.
Three additional minutes were signalled at the end of the 90 but neither threatened to score a winner in the final exchanges as the points were shared on the evening.
Both teams will now switch their attention to the Champions League as Real Madrid prepare to host Club Brugge on Tuesday, while Atletico will travel to Lokomotiv Moscow on the same night.
ATLETICO MADRID (4-4-2): Oblak; Trippier, Savic, Gimenez, Lodi (Lemar 61'); Vitolo (Correa 46'), Thomas, Saul, Koke; Felix (Llorente 70'), Costa
REAL MADRID (4-3-3): Courtois; Carvajal, Ramos, Varane, Nacho; Kroos, Casemiro, Valverde (Modric 68'); Bale, Benzema (Jovic 88'), Hazard (James 77')