Real Madrid s Brazilian forward Vinicius Junior (R) vies with Atletico Madrid s French midfielder Thomas Lemar during the Spanish League football match between Real Madrid CF and Club Atletico de Madrid at the Santiago Bernabeu stadium in Madrid, on February 25, 2023. AFP
The point provisionally reduces league leaders Barcelona's advantage on Los Blancos to seven points, but gives the Catalans the chance to expand the gap to 10 on Sunday against Almeria.
Jose Gimenez had headed Atletico in front in the second half shortly after team-mate Angel Correa had been sent off for elbowing Madrid defender Antonio Rudiger.
However 18-year-old forward Rodriguez headed Madrid level on only his second league appearance for the club.
The striker made a strong impact in a cameo against Osasuna last weekend, setting up a goal, and made an even more important contribution against Atletico.
Madrid coach Carlo Ancelotti replaced injured duo Rodrygo and David Alaba with Marco Asensio and Nacho Fernandez respectively, while Toni Kroos returned to the team.
His counterpart, Diego Simeone, was without injured World Cup winner Rodrigo de Paul and opted for youngster Pablo Barrios in the middle.
The first half was evenly contested and surprisingly pedestrian in quality, after the entertaining Copa del Rey clash these sides offered up in January, won 3-1 by Real Madrid.
Carlo Ancelotti's side also won the first league match against Atletico in October, but could not continue their domination at the Santiago Bernabeu.
Atletico defender Reinildo Mandava was forced off injured midway through the first half, appearing to hurt his knee in a tussle with Fede Valverde.
Marcos Llorente complained when he was booked for holding Vinicius Junior off with a hand to the Brazilian's face, but there was little else to set pulses rating in the opening 45 minutes.
Antoine Griezmann, in fine form in 2023, dragged a low shot narrowly wide of Thibaut Courtois's goal following good work by Yannick Carrasco after an hour.
The game tilted in the hosts' favour when Correa lashed out and hit Rudiger in the chest off the ball, with the German defender flinging himself to the floor.
Spanish news outlets had reported Atletico were unhappy with the appointment of referee Jesus Gil Manzano, who once dismissed former Rojiblancos striker Diego Costa for insulting him -- leading to an eight-match ban, and his red card for Correa will not help heal old wounds.
With the numerical advantage, Ancelotti took a risk and brought on young forward Rodriguez for defender Nacho, moving substitute Eduardo Camavinga to left back.
Seconds afterwards, Atletico took the lead.
Gimenez, who had replaced the injured Reinildo, met Griezmann's inviting free-kick with a towering header after 78 minutes to break the deadlock.
Vinicius curled an effort high over the crossbar as Madrid sought an equaliser, the Brazilian unable to recreate his sensational performance at Anfield in the 5-2 Champions League win over Liverpool midweek.
However Rodriguez intervened with five minutes left on the clock, leaping brilliantly to nod Luka Modric's corner in off the post, after Atletico had left him unmarked.
It was enough for Madrid to avoid a first home league defeat by rivals Atletico since 2016, but the result leaves the door open for Barcelona to take another step towards the title on Sunday.
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO_Sports and on Facebook at AhramOnlineSports.)