After a nervy opening, Leicester went in at the break 11-3 ahead when Genge crossed just before half-time.
And, with David Wilson in the sin-bin for Falcons, Telusa Veainu crossed to extend Leicester's lead, before Genge went over for his second from close in.
Ben Youngs then rounded off the victory with a superb break to earn his side the bonus point.
The victory opened up a four-point gap over fifth-placed Bath, who were defeated by struggling Worcester on Saturday.
Tigers, under head coach Matt O'Connor at Welford Road for the first time, had begun the day level on points with Bath, and the game looked to be heading to a 6-3 scoreline at the break after two penalties from Freddie Burns and one from Falcons' Joel Hodgson.
However, Genge was driven over from close range to put Tigers ahead at the interval, and Veainu finished off a superb left-to-right move off the back of a scrum.
Genge then followed up for his second of the match after Dan Cole had been held up just before the line and Youngs, a hopeful for this summer's British and Irish Lions tour, notched his side's fourth to wrap up a straightforward win.
Leicester head coach Matt O'Connor: "I think we have 25, 30, 40% growth in us, so it will be interesting to see how we go in the next two games.
"If we look after the ball four, five, six times better, you are talking about those numbers. We are good enough to beat anyone."
Newcastle director of rugby Dean Richards: "We never look at seventh place, we look at our performance and we were poor. Last week we were fine. We are inconsistent, which is most frustrating.
"When we are on song we play some really good rugby. The players are disappointed because we did not turn up on the day."
Leicester Tigers: Veainu; Thompstone, Roberts, Mermoz, Pietersen; Burns, B Youngs (capt); Genge, McGuigan, Cole, Barrow, Kitchener, Williams, O'Connor, Hamilton.
Replacements: T Youngs, Bateman, Balmain, Fitzgerald, Thacker, Harrison, Williams, Tait.
Newcastle Falcons: Watson; Goneva, Harris, Socino, Sinoti; Hodgson, Young; Vickers, Cooper, D Wilson, Green, Olmstead, M Wilson, Welch (capt), Latu.
Replacements: Sowrey, Harris, Welsh, Witty, Chick, Egerton, Willis, Kibirige.
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Ellis Genge's two tries helped Leicester thrash Newcastle to boost the Tigers' hopes of a top-four finish.