"We were not at our best in attack during the first half," Simeone admitted to DAZN.
"The subs gave us a bit more creativity to be able to win the game."
Simeone started his son Giuliano for the first time after some substitute appearances in recent weeks and left star summer signing Alvarez on the bench.
Atletico failed to take a single shot in a low quality first half, while Celta threatened seriously only just before the interval, when Jan Oblak saved brilliantly from Iago Aspas' drive.
The Slovenian goalkeeper made another superb stop from Borja Iglesias' close-range header after 57 minutes to keep Atletico level.
However it was Atletico who claimed victory when former Manchester City striker Alvarez, on as a substitute, stayed sharp to prod home Griezmann's deep cross.
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"(Scoring) helps for confidence, and it's always to help the team... these are three important points," said Alvarez.
"We knew that it would be a difficult game, Celta have done very well at home.
"Bit by bit we improved in the game, and in the second half we made some chances."
Earlier Ayoze Perez netted twice to claim Villarreal a 2-1 win at Espanyol, taking the Yellow Submarine fourth.
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With six goals the Spain international trails only Barcelona's Robert Lewandowski in the scoring charts by one strike.
Home fans were angry with referee Alejandro Quintero Gonzalez's lenient officiating and one threw a plastic cup which struck the official as he walked off at full-time.
Real Betis could only secure a 1-1 draw at bottom of the table Las Palmas.
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Barcelona lead the league after their 1-0 win over Getafe on Wednesday with Real Madrid second, beating Alaves 3-2 on Tuesday.
(Except for the headline, this story has not been edited by staff and is published from a syndicated feed.)