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

The Russian, 29, tested positive for meldonium at January's Australian Open.
When asked if Sharapova would play any more tournaments, Shamil Tarpishchev told R-Sport news agency it was "very doubtful" and added the five-time Grand Slam winner was in a "bad situation".
In March, Sharapova said she was "determined to play tennis again".
The International Tennis Federation (ITF) provisionally suspended Sharapova on 12 March.
She is waiting to hear the full extent of her punishment, which could be as much as a four-year ban, although experts say a six-month or 12-month suspension is more likely.
That is because the World Anti-Doping Association (Wada) admitted in April that scientists were unsure how long meldonium stayed in the system.
It even suggested athletes who tested positive for the substance before 1 March could avoid bans, provided they had stopped taking it before 1 January.
However, Sharapova has already admitted she continued taking meldonium past that date, saying she was unaware it had been added to the banned list as she knew it by another name - mildronate.
We've launched a new BBC Sport newsletter ahead of the Euros and Olympics, bringing all the best stories, features and video right to your inbox. You can sign up here.
Tom Briscoe's hat-trick helped Leeds to an easy Qualifiers victory over London.
[NEXT_CONCEPT]
Former world number one Maria Sharapova may never play again following her failed drugs test, the president of the Russian Tennis Federation says.