Damsgaard joined Sampdoria in July 2020, scoring twice in 49 games.
In the Euro 2020 semi-final last summer, the 22-year-old scored a free kick against England, but Denmark lost 2-1 after extra time.
Damsgaard joins Brentford for an unspecified sum of about 15 million euros (£12.7 million).
Thomas Frank, the manager of Brentford, claimed Mikkel was “another fantastic Brentford signing. He is a young talent with a lot of growth potential.
Frank said he had had a bad year of injuries, but we have already seen him as a very skilled player.
He is a player who will fit into our locker room, which is important. He has a good character, wants to put in a lot of effort, work for the team, and wants to grow.
The player will train this week before Brentford’s Premier League match against Manchester United on Saturday, subject to international clearance.
In their first league matchup against Leicester on Sunday, the Bees rallied from a 2-0 deficit to draw.
Damsgaard joins Ben Mee, Aaron Hickey, Keane Lewis-Potter, and Thomas Strakosha as Brentford’s fifth summer additions.
He has made 16 appearances for his nation, scoring four goals, and in a season marred by injuries, he played 11 times for Sampdoria.
Before joining Sampdoria, Damsgaard spent three seasons in the Danish club FC Nordsjaelland’s youth, where he made his debut at 17.