The 21-year-old, who appeared as a substitute in Portugal's 1-0 loss to Belgium at Euro 2020, has been in pain since December when he fractured his foot.
Morata returned to Juventus in September having previously played for the Turin giants under Massimiliano Allegri between 2014 and 2016.
Fans and players later celebrated together outside the stadium in Valladolid, with more supporters rejoicing back in Madrid.
Suarez scored in the 58th minute after Angel Correa had equalised Oscar Plano's opener for Valladolid.
Real Madrid will play against Villarreal in their last game of the season, and Zinedine Zidane's side needs to win this match in order to keep their hopes alive of winning the league.
Recent years have been harder. Atletico were playing well in the previous seasons, they were never really at the top as it was dominated by Real Madrid and Barcelona, says Luis Garcia
Atletico leads defending champion Real Madrid by two points with two matches remaining and will win the title if it beats Osasuna at home and Madrid fails to win at Athletic Bilbao.
Atletico Madrid have a chance to win La Liga 2020-21 if they win all the three games that are left in the Spanish competition.
Atlético played well at the Camp Nou but couldn't capitalize on its chances, with Luis Suárez unable to break through with a goal against his former club.
Atletico were 10 points clear at the start of February but six wins in 14 games since then have seen that lead reduced to two.
The moves by Atlético and Inter came a day after the six Premier League clubs involved in the project gave up on the controversial breakaway competition.
After two straight games without a victory, Real Madrid is now only one point behind and Barcelona is two points behind with eight games left.
Despite the defeat, Atletico Madrid holds the top spot on the La Liga table with 66 points from 29 games.
Atletico Madrid's Luis Suarez scored his 500th career goal as he guided his side to a 1-0 victory over Alaves in LaLiga.
Hakim Ziyech and Emerson scored one goal each in the match to power the Premier League side into the quarter-finals of the competition.