The agreement includes an initial payment of £65m, with an additional £6m in performance-related add-ons.
Mbeumo, who netted 20 Premier League goals last season, had made it clear that Old Trafford was his preferred destination, despite interest from Newcastle and Tottenham.
This transfer represents a club record sale for Brentford and will be the most significant acquisition under Ruben Amorim's tenure at United.
The deal was secured after weeks of negotiations, with United submitting a third offer to meet Brentford's valuation, which was benchmarked against other recent player sales.