The 27-year-old has signed a two-and-a-half year contract following a successful loan period at the club.
Tudgay has scored three goals in as many starts, including two in the 5-2 victory over his former club Derby County at the end of December.
Meanwhile, Mark Byrne, Julian Bennett and Joe Garner have returned to the club after their loan deals expired.
Byrne scored four goals in 15 matches for Barnet, but has recently been out of action with a knee injury.
Bennett returns from Crystal Palace following one goal in 11 starts and three substitute appearances at Selhurst Park.
And Garner arrives back at The City Ground after 19 games for Huddersfield, 11 as a starter and eight from the substitutes bench.
Source: BBC Sport