It is the second successive new deal for the Clarets, with Ashley Barnes signing earlier in the week.
Woods has committed his future to the club until the summer of 2023.
The New Zealand international has scored 24 Premier League goals since signing from Leeds United in August 2017.
"It's great news for both of us [him and Barnes]. It's always nice to be recognised and it's fantastic that it's over the line and sealed," said the 27-year-old after penning his new deal.
"It's nice to have the two years under my belt and to be settled and happy where I am and now it's about pushing forward and doing even better.
"It feels exactly like home. It's a nice base and it's a nice settled place for me.
"I haven't really settled down previously at clubs. I think two-and-a-half is the maximum I've been at a club so hopefully to be here for four, five, six, seven years would be lovely.
"I am very happy to have it signed and looking forward to the future."