Broncos led through Rhys Williams before Briscoe and Liam Sutcliffe posted two quick scores for the Rhinos.
James Cunningham put London back ahead but tries from Briscoe and James Segeyaro made it 20-12 at the break.
Jordan Lilley, Matty Achurch and Kallum Watkins extended Leeds' lead, and despite tries from Nick Slyney, Iliess Macani and Elliot Kear either side of Briscoe's third, Leeds secured victory.
Leeds maintained their 100% record in the Qualifiers, having previously recorded victories over Featherstone and Hull KR.
But there was more injury woe for the Rhinos as captain Danny McGuire limped off with an ankle injury in the first half.
London Broncos head coach Andrew Henderson: "I am very proud of the efforts of the guys.
"We put ourselves under pressure at times and we had to absorb that and I was pleased to see us finish the game off strongly, scoring some points at the back end of the game to make the score more respectable."
Leeds head coach Brian McDermott: "There were spells where I thought we were really good and some other spells in the game where I thought we were very off.
"We made some periods of the game look very difficult at times with our lack of energy, but also London gave it a good crack against us and they didn't fall off like perhaps Featherstone did last week."
London Broncos: Walker; Williams, Hellewell, Kear, Macani; Soward, Barthau; Slyney, Cunningham, Ioane, Harrison, Garside, Bussey.
Replacements: Ackers, Thackray, Battye, Offerdahl.
Leeds Rhinos: Sutcliffe; Briscoe, Watkins, Keinhorst, Hall; Moon, McGuire; Galloway, Segeyaro, Garbutt, Ferres, Ablett, Jones-Buchanan.
Replacements: Cuthbertson, Achurch, Mullally, Lilley.
Referee: Chris Campbell
Tom Briscoe's hat-trick helped Leeds to an easy Qualifiers victory over London.