Dowd was selected by the Los Angeles Kings in the seventh round (No. 198) in the 2009 NHL Draft after playing for Wenatchee of the North American Hockey League. The following season, he played for Indiana of the United States Hockey League and had 39 points (16 goals, 23 assists) in 46 games.

Dowd played the next four seasons as a forward for St. Cloud State University, where he had 121 points (52 goals, 69 assists) in 155 games. He was named to the All-National Collegiate Hockey Conference First Team and was a finalist for the 2014 Hobey Baker Award, awarded to the top player in NCAA Division I hockey.

He signed a one-year, entry-level contract with the Kings on April 1, 2014.

Dowd played most the following two seasons with Manchester and Ontario of the American Hockey League. In 2015-16, he had 48 points (14 goals, 34 assists) in 58 games for Ontario and was called up by the Kings. He made his NHL debut March 22, 2016, a 2-1 loss to the Minnesota Wild, and was held without a point in five games before being returned to Ontario.

Dowd made the Kings at training camp in the fall of 2016 and had 22 points (six goals, 16 assists) in 70 games for Los Angeles, playing mainly on the third and fourth lines. He had one assist in 16 games with the Kings in 2017-18 before being traded to the Vancouver Canucks on Dec. 8, 2017, for defenseman Jordan Subban.

In 40 games with the Canucks, Dowd had three points, all goals.

The Washington Capitals signed Dowd as a free agent July 1, 2018. He since was named the recipient of the organization's second-annual Caps Care Community Award in honor of his efforts in the community during the 2024-25 season, including through his participation in his own player program, Dowd’s Crowd. Dowd also became what Capitals general manager Chris Patrick called "an integral member of our team, both on the ice and in our leadership group" to earn a two-year, $6 million contract he signed with Washington on April 16, 2025.

Dowd was traded to the Vegas Golden Knights on March 5, 2026. His short-handed ice time per game of 2:40 was the highest among Washington's forwards at the time of the deal.