Woods' first goal then proved the final straw for hundreds of home fans, who charged along the concourse to gather underneath the directors' box to chant "sack the board and "you destroyed our club".
Wood told Match of the Day he wondered if the game was going to stopped. "It's a tough situation, I can sympathise with the fans. They just want to see their team win. It's tough for them at the moment but they need to stick with the players," he said.
"You don't want that to happen on the pitch, it creates problems and safety issues. We were fine, they were just voicing their frustration."
The fan behaviour took some of the gloss off a brilliant Wood cameo which kept Burnley seventh in the standings.
It was the 26-year-old's first double for the club, lifting him to an excellent strike rate of nine goals in 22 appearances for Burnley.
He scored off the bench in last week's 2-1 win over Everton, a match which ended two months on the sideline recovering from a knee injury.
Wood was denied a chance to lock horns with All Whites team-mate Winston Reid.
West Ham centre back Reid's season ended prematurely last weekend when he was forced off the field with a serious knee injury.
