The loss left toothless Chelsea 11th in the Premier League table, stumbling towards possibly their worst campaign since the 1993/94 season - when they finished 14th under Glenn Hoddle - despite their new US owners' huge outlay on players.
Brentford went ahead when defender Mathias Jorgensen headed on a corner at the near post and the ball struck Azpilicueta and went in after 37 minutes, before the visitors had even had a shot of their own on target.
Lampard - Chelsea's third coach this season after Thomas Tuchel and Graham Potter were fired - brought on winger Mykhailo Mudryk and striker Pierre-Emerick Aubameyang at halftime as he sought to end a goal drought that has now seen the London side score only once in seven games in all competitions.
But it was substitute Mbeumo who showed the Blues how to do it in the 78th minute when he drifted in from the right and his fiercely hit shot was deflected beyond Kepa Arrizabalaga.
The three points pushed Brentford up a spot to ninth in the league, eight points ahead of their big-spending west London neighbours for whom this season cannot finish quickly enough.
Media reports said Chelsea were close to hiring former Tottenham Hotspur manager Mauricio Pochettino as coach from next season.
"This is not a sob story but everything is going against us," Lampard told BT Sport, saying Chelsea had controlled much of the game but failed to create clear chances, something he attributed to a lack of confidence.
"We would love a magic wand, a magic moment. It doesn’t come in football because the Premier League moves so fast, it’s competitive," the former midfielder who is Chelsea's all-time top scorer, said. "So we will have to fight for our moment."
Brentford coach Thomas Frank was able to celebrate surpassing his side's points tally for all of last season - their first in the Premier League - with Wednesday's win.
"You can't take that for granted. That takes a lot of hours on the training pitch, in the offices, over a year to be in a position where we are now," he said.
