Puig flashed the middle finger on both hands at fans sitting behind home plate at Progressive Field after hitting a home run in the second inning.
Baseball executive Joe Torre announced the one-game penalty and undisclosed fine on Wednesday.
Puig is appealing the decision. He was in Wednesday's night lineup and batting ninth for the second game of the three-game series against the Indians.
The mercurial player said he "stooped" to the level of fans who he said were heckling him in the on-deck circle. Puig said that the group of fans kept talking to him after the home run.
Dodgers manager Dave Roberts spoke with Puig about the incident on Wednesday. The manager addressed the media before the suspension was announced.
"I made it clear I was not pleased with what he did," Roberts said. "He understands how I felt -- and he won't do it again. To have us talking about obscene gestures, I don't want to do that and he doesn't want to do that."
Roberts said Puig regretted the incident.
"It's something that he wishes didn't happen," Roberts said. "He didn't appreciate what was being said, and he reacted."
The suspension is the latest incident involving Puig, who has tried the Dodgers' patience with previous behavior.
Puig tallied two hits on Tuesday in the Dodgers' 7-5 win over the Indians.
Roberts played in the majors for 10 years, and he realizes there's a give-and-take with fans.
"You either have fun with it, which is what I tried to do, or maybe you go back a little with them to try and win them over," Roberts said. "But sometimes, when it is in bad taste, you get emotional."