The 19-year-old will stay on loan at Burton until the end of the season.
Lucas told the club's official website: "I'm really looking forward to it. I can't wait to play some games and gain some more experience.
"I can't wait to test myself against experienced players in League Two and sample the pressure that competitive football brings. I had a taste of league football last season against Sheffield United. I really enjoyed it and it spurred me on for more.
"The manager called me into his office and told me about the possibility of going out on loan. He said it was a good opportunity for me, especially as Burton try to play the game the right way and are in with a chance of promotion. Hopefully I can help them achieve that goal.
"To be honest, I had no hesitation in accepting the opportunity because I've been hoping to go out on loan for a while. The new reserve league has been good for me this season but I felt I needed to take the next step."
Burton boss Paul Peschisolido said: "He's a lad we've watched a few times and he's been in and around Swansea's first team, he trains with the first team. He's got good energy and he's an excellent footballer. He's a young lad and I know Brendan [Rodgers] is very keen for him to go out and play games.
"He's got a great range of everything, he keeps the ball. I don't want to get too carried away because he's a 19-year-old boy but what we have to do is get him on the ball. Swansea have players who are comfortable on the ball and that's how I want us to play."