Scott Parker’s men went into the game needing a win to avoid the drop, but yet again struggled in the final third and were unable to find the breakthrough.
Fulham knew they would have to surpass their great escape from relegation in 2007-08, in which they won their last three matches, but their 12th home league defeat of the campaign ensured they would be playing Sky Bet Championship football next season.
Goals have been hard to come by for Parker’s team, with just 25 in the Premier League, and despite dominating possession they were unable to add to their tally.
Fulham started brightly and committed numbers forward. Frank Anguissa had the first opportunity of the match, but his shot was straight at Nick Pope inside the opening seven minutes.
The visitors had the ball in the back of the net when Wood tapped the ball past Alphonse Areola, but it was disallowed.
The France international was called on again from a corner to bundle the ball behind on the goal line, but Burnley looked to be getting closer.
The Clarets were then rewarded for their efforts when Westwood was picked out at the back post by Matej Vydra and he made no mistakes slotting the ball home.
With half-time approaching Fulham’s challenge was made harder, and their fate was all-but sealed when Wood struck the ball expertly into the top corner from distance.
Fulham went into the second half needing three goals without reply and, having managed just nine goals at home all season in the Premier League, it looked an unlikely scenario.
Two penalty appeals were heard from the Fulham bench, first Lookman, who stumbled in the box, but a louder shout was when Ivan Cavaleiro went down under a heavy challenge, but neither were given.
Areola could have been awarded a red card himself, after coming out of his area to make a crucial challenge, but the ball appeared to strike his hand and VAR decided not to intervene.
Fulham kept up the pressure until the end, with a makeshift back three to commit more attacking players.
Anguissa came closest when he rattled the bar from the edge of the area, but Parker’s side could not get on the scoresheet and their Premier League stay came to an end.