The Bees experienced a disappointing evening as they fell to their ninth defeat in their last 11 Premier League matches.
England international Jarrod Bowen scored twice in the opening seven minutes to put Brentford on the back foot at the London Stadium.
Although Neal Maupay halved the deficit in the 13th minute, Bowen went on to complete his hat-trick after the half-time break, before Emerson Palmieri added a fourth to West Ham's tally.
Yoane Wissa came off the bench to find the net in the closing stages, but it did little to lift the mood as the Bees slumped to a third consecutive defeat.
Brentford's frustrating evening was compounded by the fact that Mee limped off in the 61st minute when West Ham held a narrow 2-1 lead.
The 34-year-old, who was replaced by Nathan Collins, is now a fitness concern ahead of Saturday's meeting with Brentford's West London neighbours Chelsea.
While Thomas Frank was unable to provide too many details on Mee's injury, he confirmed the centre-back was forced off with an ankle issue.
"It's something to the ankle," Frank told reporters at the post-match press conference. "I don't know exactly yet. We will assess him tomorrow."
Frank will be hoping for a positive diagnosis, with Mee regarded as one of the key players in the Brentford squad, having started each of the last 13 Premier League games that he has been eligible for.
The experienced defender has made a total of 18 competitive appearances this season and has chipped in with a couple of goals.
Brentford are already having to cope without the services of Ethan Pinnock, Aaron Hickey, Rico Henry, Josh Dasilva, Kevin Schade and Bryan Mbeumo.
Having picked up 25 points from their 26 Premier League matches, the Bees are currently sitting in 16th place and five points above the drop zone.