Bowe, who is eligible to practice with the Chiefs throughout the preseason, will miss the opener against the Tennessee Titans on Sept. 7 at Arrowhead Stadium. He also can play in the Chiefs' three remaining preseason games, including Sunday night's contest against the Panthers in Charlotte.
"I am extremely sorry that I can't be on the field for the first game of the 2014 season because I made an error in judgment," Bowe said in a statement. "I take responsibility for my mistake, and it will not happen again. I hope that my teammates, coaches and Chiefs fans accept my sincere apology.
"I will be supporting my teammates in Week 1, and I cannot wait to get back on the field to help us win in Week 2 and beyond as we work towards our goal of winning the Super Bowl."
The Chiefs said they are "disappointed" by the situation.
"Dwayne has been given a one-game suspension by the National Football League," the team said in a statement. "We are disappointed in the situation and will miss him on the field, but we support the league's decision. The club will have no further comment on the matter."
Bowe was arrested in November by police in Riverside, Missouri, for possession of marijuana. He pleaded guilty in April to lesser charges of littering and defective equipment. The marijuana charge was dismissed.
The passenger in Bowe's car the night of the arrest told police the marijuana belonged to him and that Bowe did not know it was in the car.
The Chiefs have a variety of candidates to replace Bowe, who has led the team in receptions in each of his seven seasons. They include Junior Hemingway, A.J. Jenkins, Kyle Williams and rookie Albert Wilson.
Those players have a combined total of 68 NFL catches between them. Bowe has 472.
Bowe also was suspended without pay for four games in 2009 after violating the league's policy against performance-enhancing substances.
ESPN NFL Insider Adam Schefter contributed to this report.