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Godin pulled a muscle in the 64th minute on Saturday as Inaki Williams gave Bilbao a 2-1 lead at the Wanda Metropolitano, but Atletico's captain stayed on.
Dortmund handed Atletico their heaviest defeat under Diego Simeone with a 4-0 victory 13 days ago but this contest was almost a mirror image, the hosts in control from start to finish.
Griezmann's bending effort in the second half looked likely to prove the difference in a contest short on chances at Butarque but Guido Carrillo's 82nd-minute equaliser earned Leganes a deserved point
Diego Simeone declined to comment on the situation of Gelson Martins, who has made only three appearances with the Rojiblancos since joining the club over the summer.
Atletico Madrid lead second-placed Barcelona by one point, with Real four further adrift, with the two giants to face off at the Camp Nou on Sunday.
Champions League: Facing Atletico Madrid is like toothache for Borussia Dortmund goal-ace Paco Alcacer
The 25-year-old, on loan from Barcelona, has needed just 106 minutes to score seven Bundesliga goals this season -- the best goal-rate in the league's history.
Atletico sat seven points behind Real Madrid and Barcelona after winning only one of their opening four league games but as their rivals have stumbled, Diego Simeone's side have sprung into form.
It means Real second, level on points with Barcelona, while Atletico are just two behind in fourth. Sevilla are third.
Some had put a line through Atletico a month ago but they will move above Real with a win - and could even go top if Barca's dip continues at home to Bilbao.
One victory from their opening four league games left Atletico seven points adrift of Barcelona a fortnight ago but three on the spin have now steered Diego Simeone's side back on track.
The visitors had fallen behind against the run of play in the 18th minute when winger Samuel Grandsir forced the ball home in a goalmouth scramble after good work from Radamel Falcao.
Griezmann had teased Ramos after Atletico beat Real Madrid 4-2 to win the UEFA Super Cup in August.
In the 2016-2017 season Monaco beat Manchester City and Borussia Dortmund before losing to Juventus in the semi-finals, with club owner Dmitry Rybolovlev then selling a raft of stars.
Atletico Madrid star Antoine Griezmann says he still can't fathom why he was not voted one of the world's top three players by FIFA.
For Griezmann, who helped Atletico to Europa League glory as well as playing a pivotal role in France's World Cup win, FIFA's choice was unwarranted.