Briscoe topped his previous high of 20 last December at UCLA behind 9-of-13 shooting that helped Kentucky (2-0) rally from a seven-point first half deficit to build a 44-35 halftime advantage.
Freshmen and fellow guards Fox and Monk combined to shoot 11 of 24 en route to career scoring bests in their short careers, while 6-foot-10 newcomer Bam Adebayo had 14 points and 11 rebounds a game after a foul-plagued debut on Friday.
Kentucky shot 51 percent from the field.
Kassius Robertson had 16 points, Jermaine Crumpton 13 and Phil Valenti 12 for Canisius (0-1) in the coaching debut of Reggie Witherspoon. The Golden Griffins initially pushed the pace and had Kentucky reacting but had no answer for the Wildcats’ backcourt.