Rory McIlroy -- FILLS TROPHY WITH JAGER ... After British Open Victory

The Claret Jug's probably nursing a pretty nasty syrupy hangover this morning -- 'cause Rory McIlroy filled it up with JAGERMEISTER after his British Open victory.

It appears Rory's mom was part of the raging after-party ... and stabilized the Jug while one of Rory's pals loaded it up with the Jager like it was a college sophomore in 2003.

Rory's victory drink was about $39,980 cheaper than Phil Mickelson's ... who recently said he filled the Jug with a $40k bottle of wine after he won the Open last year.

FYI -- seems Jager's his drink of choice ... Rory filled the U.S. Open trophy with the black sauce after winning that tourney back in 2011.