The 22-year-old has joined compatriot Thomas Frank's side on a five-year contract lasting until the summer of 2027.
Damsgaard joined Sampdoria from Danish side Nordsjaelland in the summer of 2020 and made 49 appearances for the Serie A side scoring two goals.
He was one of the standout players for Denmark in their run to the semi-finals of last summer's European championships, scoring the tournament's only free kick goal against England.
Injury restricted Damsgaard to just 11 Serie A appearances last season but he is expected to provide Brentford with creativity in the middle of the park following compatriot Christian Eriksen's departure for Manchester United this summer.
Speaking to the club's official website, head coach Thomas Frank said: "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, we have the staff here that have experience in doing that and we are looking forward to working with him.
"Mikkel is a player that has shown his promise at FC Nordsjælland and Sampdoria as well as in the European Championships last summer, when he had a fantastic tournament for Denmark. He has been unlucky with injuries in the past year, but we have already seen that he is an extremely skillful player, who can run with the ball and move it well.
"We think Mikkel can play as one of our 'eights' in a 4-3-3 system or as a 'ten'. He can also play on the wing. We will use him in the way we think we can get the best out of his offensive abilities. Importantly, he is a player that will fit into our dressing room. He is a great character, he wants to work hard, he wants to work for the team, and he wants to develop."
Damsgaard could make his debut against Man Utd this weekend.
The Bees started their season with a 2-2 draw away at Leicester City last weekend.
