Boro beat off stiff competition to land the Republic of Ireland international, 21, who was also wanted by a number of rival Championship clubs.
Connolly, who has represented his country at all levels from Under-17 and has eight senior caps, has scored twice this season and his most recent top flight appearance was in December against Wolves.
He has made over 50 appearances for Albion and netted eight times for the south coast club, scoring twice on his top-flight debut against Spurs in 2019.
On his new arrival, Wilder said: “I’m delighted. It’s an area of the pitch we need to improve in terms of competition for places.
“He’s a young player, he’s played in the Premier League a number of times, and he’s from a technical football club. He’s also played for his country.
“He gives us pace, can finish at the top of the pitch and gives us competition for the players around him."