Jonny Lee Miller -- I Fight Crime On and Off Screen!

Jonny Lee Miller is taking his new gig as TV's Sherlock Holmes seriously ... 'cause he took it upon himself to fight some crime in L.A. on Friday ... and WON!!!

Law enforcement sources tell TMZ ... the "Elementary" star noticed some guy messing with his mailbox and decided to take matters into his own hands, by chasing the dude down.

Highly trained to "act" like he fights crime, we're told Miller confronted the mail thief -- the two exchanged nasty words, then an actual scuffle broke out as Miller tried to stop the guy from stealing other people's mail.

Sources tell us Miller called 911 while still exchanging blows with the alleged criminal ... and when the cops arrived, they arrested the man on the spot for mail theft and recovered the stolen mail.