The New Zealand international is the latest signing for the club after they were taken over by Saudi Arabia's Public Investment Fund, with Kieran Trippier joining the squad last week from Atletico Madrid.
"Chris is an important signing for us at a crucial time and I'm very pleased we have been able to move quickly and bring him here," Newcastle coach Eddie Howe said.
"He is a very dangerous attacking threat, has a physicality and character that I really like, and he has vast experience in the Premier League. He will be a great fit for us."
Wood, 30, has scored three goals in 21 matches for Burnley, who are level on points with Newcastle put a place above the north-east side in 18th place though have played two matches fewer, this season.
He said: "This is a very exciting opportunity for me and I'm delighted to be a Newcastle United player.
"I have played at St. James' Park a few times and it's an amazing place, so to have the chance to represent this club with those incredible supporters behind me and my teammates is very special.
"The club and manager have been first class during this process, and I really can't wait to get onto the pitch and repay the faith they have in me, especially with a vital part of the season ahead of us."
Newcastle sought to bolster their attack after news that Callum Wilson would be out for up to eight weeks with a calf injury. The club are second from bottom with 11 points from 19 games. Watford, who occupy 17th spot just above the relegation zone, are on 13 points after 18 games.
Information from ESPN's Tom Hamilton and Reuters was used in this report.