The 27-year-old Argentine will go straight into the squad for Tuesday night's visit of Colchester.
Charlton have an option to make the deal permanent at the end of the campaign.
Bessone's last appearance for Leeds came in a 5-2 defeat at Barnsley in September.
Bessone was signed by Leeds boss Simon Grayson last summer but has found first-team opportunities hard to come by.
Fellow left-back Matt Fry's loan spell at The Valley from West Ham is due to expire next weekend, although that could yet be extended.