Bournemouth were undone by two set-piece situations, having initially taken the lead in the 18th minute through an own goal from Vitaly Janelt. However, Brentford responded with goals from Yoane Wissa and Norgaard, sealing a fifth consecutive away win for the Bees.
Wissa equalised in the 30th minute, heading in from a corner, before Norgaard seized on a long throw-in in the 71st minute, which bounced invitingly in the home side's penalty area.
Brentford climbed to 11th in the 20-team table, while the loss dealt a blow to Bournemouth's European aspirations, as they remain in ninth after a third consecutive home defeat. The result was particularly damaging, as a win would have lifted Bournemouth to sixth.
The match began with Bournemouth taking the lead in the 18th minute. After a ball was switched to the left wing, Antoine Semenyo played a pass to the overlapping Milos Kerkez, who squared the ball across the penalty area. The ball struck Janelt for an unfortunate own goal. It was a deserved lead for the hosts after a strong start, but Brentford quickly levelled the score in the 30th minute when Mbeumo's in-swinging corner was met by Wissa, who outjumped the home defence. Goalkeeper Kepa Arrizabalaga misjudged his punch and was caught out of position.
Less than a minute later, Mbeumo came close to putting Brentford ahead but his shot narrowly missed the target following a swift counter-attack. Kevin Schade's pace set up another chance just before halftime, but his pass inside to Wissa didn't find its mark.
Bournemouth almost regained the lead two minutes into the second half when a precise cross from Justin Kluivert found Semenyo unchallenged in the box. The forward's header, however, struck the crossbar.
There were a few more half-chances for Bournemouth before Brentford struck again. Schade's long throw-in was allowed to bounce through to Norgaard, who fired it home from close range in the 71st minute, securing all three points for the visitors.
This victory marked Brentford's first-ever five-game winning streak on the road in the Premier League.