The Northern Ireland international has joined the Clarets on a free transfer having been without a club since leaving Reading back in January.
The 32-year-old former Southampton and Fulham player has penned a short-term deal until the end of the season.
Baird can play both in central midfield and at full-back. He featured for Reading in the Royals’ 2-1 defeat at Turf Moor earlier in the season.
He will now link up with Sean Dyche’s squad ahead of Saturday’s game against Charlton Athletic.
Upon his arrival at Turf Moor, Baird said: “It feels great! It’s just nice to be part a team that is playing so well at the moment.
I don’t mind where I play, as long as I’m getting some games. I think that’s the most important thing.
I know it’s not the biggest squad, and there’s been a few injuries. But I’ve spoken to the manager and he really wanted me here and I wanted to come to try and help the team finish the season as strong as possible.
I’ve been doing a lot of training at Reading to keep myself fit. I’m fully aware of the situation at the Right back area, it’s up to manager if he wants me to play, but if he does I’m ready to play.”