An abysmal performance saw the Bees take lead after only 10 minutes thanks to a howler from David de Gea, who let the ball slip through his palms.
De Gea was at fault again nine minutes later, selling Christian Eriksen short with a pass and allowing Mathias Jensen to nip in and double the lead.
Brentford were 3-0 up within half an hour when Ben Mee bundled the ball home at the far post from a corner, easily beating centre-back Lisandro Martinez in the air despite losing his balance in the process.
It was 4-0 five minutes later when Ivan Toney teed up Bryan Mbeumo to finish past De Gea, to confirm United's worst-ever half-time scoreline and facing a record away Premier League defeat.
Heading into the season, Ten Hag was 33/1 to be the first manager sacked but after losing to Brighton last week and seeing a first-half thumping drop United to the bottom of the table, he is now 13/2 to get the boot.
Odds for first Premier League manager sacked (via Betfair)
Ralph Hassenhuttl 2/1
Frank Lampard 4/1
Erik ten Hag 13/2
Brendan Rodgers 9/1
Stephen Gerrard 9/1
Manchester United face Liverpool next week.
Additional reporting by Press Association.