Ryan Hall nearly scored an opener for the Hatters after just 15 seconds at Victoria Road, but he poked wide after Scott Doe’s poor header.
Paul Benson came close to scoring in the seventh minute, but Joe Widdowson headed off the line to prevent the former Dagger netting against his former club.
The hosts slowly worked themselves back into the encounter and Jamie Cureton fired into the side-netting before Hatters were reduced to 10 men.
Benson and Ian Gayle contested a bouncing ball and the Luton forward challenged with his foot, whilst the Daggers defender went with his head.
Referee Darren Sheldrake was left with no choice but to show a red card to Benson in the 38th minute after he caught Gayle high in the face.
Burnett’s team started the second period strongly and Cureton should have made more of two opportunities that came his way.
Elliot Lee went close from a free-kick for the visitors, but Mark Cousins tipped wide and it finished goalless.