Bedford-born Burgess, 27, moves to Kenilworth Road after being released by Rushden & Diamonds.
Hatters boss Mick Harford told the club's official website: "Andy has got the experience of the Conference. He has won the title with Rushden and played 50 games for Oxford.
"He has a great range of passing and he will come here with the experience we need to play in this division.
"He is a big Luton fan and his family have a big affinity with the club. I know when he plays he will run that extra yard for the football club."