Atletico Madrid had the better of the chances early on, but it was Bayern that opened the scoring when Coman settled a perfect Joshua Kimmich ball over the visitors' back line and beat Jan Oblak with his next touch to make it 1-0.
A poor giveaway by Atleti led to Bayern's second with Coman finding space in the area and squaring to Leon Goretzka, who volleyed home from a tight angle to double the hosts' lead ahead of half-time.
Joao Felix looked to have cut the deficit in half just after the restart, but his well-taken volley that beat Manuel Neuer was chalked off after VAR ruled that Luis Suarez had blocked the Bayern keeper's view from an offside position.
Corentin Tolisso effectively put the game away in the 66th minute when he collected a deflected clearance and hit a thunderous strike past Oblak into the top right corner from 25 yards out.
Coman then popped up again for his second on the counter-attack, blazing his way from midfield and through the Atleti defence to beat Oblak yet again for Bayern's fourth goal of the match.
The Germans were without their in-form winger Serge Gnabry, who tested positive for COVID-19 on Tuesday.
The result sees Bayern Munich top their group and extend their Champions League record winning streak to 12 games dating back to last season.
Bayern Munich coach Hansi Flick praised his team's efficiency following the comfortable win.
"Our game was just good tonight," Flick told reporters. "It was a tough task but we mastered it. Our efficiency was good today and I am very satisfied. It is very important to win that first game."
Bayern won the Champions League title in August, one of five trophies they won in 2020, and had a short preseason preparation but Flick said his players fought for every ball against Atletico.
"We were physically present. That was important, to be there, to win those one-on-one situations. We had a match plan and we applied it, playing out four beautiful goals."
"We can be satisfied with it and that is what matters. We really, really wanted to win the first game today," Flick said.
Atletico boss Diego Simeone said he was unsure what caused the "heavy defeat" after the match.
"I don't know if we made big mistakes or they were down to their virtues," Simeone said. "When you lose the ball they get forward well. I'll focus on the team's attitude, we looked for a goal and it could have come if we had been clearer in our finishing. We had the offside goal. The opponent being clinical made the difference."
He added that the loss is one that will not easily be forgotten.
"The bad taste of the defeat will stay with me but I'll focus on a lot of positive things and as I'm an optimist I'll focus on the things that will make us better in the future," he said.