The 22-year-old has signed a five-year contract with the Bees and will link up with the squad in training ahead of Saturday's home clash against Manchester United. The move is subject to international clearance.
Brentford head coach Thomas Frank told the club's website: "Mikkel is, in my opinion, another classic Brentford signing.
"He is a young player with big potential to develop further. We know we can help Mikkel reach his highest level.
"Mikkel is capable of creating and scoring goals and he has the ability to run with the ball past players. That is a great asset to have in the squad."
Damsgaard came through the youth ranks at FC Nordsjaelland and made his debut in the Danish top flight at the age of 17.
He moved to Sampdoria two years ago and made 35 appearances for the Serie A side in his first season.
He went on to catch the eye at the delayed Euro 2020 tournament with Denmark, for whom he has 16 caps, but his 2021-22 campaign was disrupted by injury.