The hosts were keen to start quickly with Milos Kerkez having a shot saved before Evanilson and Ryan Christie each saw an effort blocked.
Dean Huijsen then strode forward and smashed a shot that was deflected wide by Nathan Collins, and Evanilson dinked an attempt off-target from a tight angle.
The Cherries eventually made the breakthrough in the 17th minute, when Kerkez latched onto Antoine Semenyo’s pass and fizzed in a ball that bounced off the visitors’ Vitaly Janelt and into the net.
Kevin Schade was found twice in the hosts’ box shortly after, but he saw a tame header routinely saved by Kepa Arrizabalaga and was then let down by a poor touch that allowed Illya Zabarnyi to clear.
The visitors were still pushing forward and duly equalised on the half-hour mark, as Yoane Wissa met Bryan Mbeumo’s corner ahead of Kepa and guided his header inside the post.
Mbeumo and Schade both came agonisingly close to adding a quickfire second, before Evanilson’s bouncing attempt was tipped away by Mark Flekken at the other end.
Much like the first half, Bournemouth started quickly in the second period but Semenyo’s header from Justin Kluivert’s cross rattled the crossbar on this occasion.
Both teams continued to compete well, but there was a stage where goalscoring opportunities came at a premium.
Evanilson teed up a Marcus Tavernier shot that marginally missed the top corner, but it was Brentford who struck next, as Schade’s long throw wasn’t dealt with and Christian Norgaard reacted to finish clinically and complete the turnaround.
David Brooks met a pinpoint Luis Sinisterra cross well but fired straight at Flekken, who also denied Tavernier and Semenyo to ensure Brentford earned their fifth consecutive away win to go up a place to 11th in the Premier League.
They are also only three points behind Bournemouth, who haven’t won any of their last five matches across all competitions inside 90 minutes - although that run includes their penalty shootout win over Wolves in the FA Cup.
Flashscore Man of the Match: Christian Norgaard (Brentford)