Already leading 4-1 from the first leg last week, Leicester was never in danger once Harvey Barnes opened the scoring after just two minutes.
James Maddison added two more in a four-minute spell in the second half before Stephen Odey secured a consolation.
"We got a good result and, over two legs, we scored seven goals," said Leicester boss Brendan Rodgers, who has been under pressure for his side's faltering Premier League form.
"That's very, very pleasing, so we just have to focus and look forward to our next game.”
Celtic were convincingly dumped out 5-1 on aggregate as Bodo dominated in a 2-0 win in Arctic conditions.
Visiting manager Ange Postecoglou hinted he believed the tie may have already been lost in Glasgow last week as he made a much-changed side with a series of key players left out.
Bodo took full advantage in a blistering opening as Ola Solbakken opened the scoring after just nine minutes.
Hugo Vetlesen then calmly steered home a second 21 minutes from time at the end of a well-worked move to keep alive the European dreams of a side from a town with a population of just 50 000.
"It's fair to say that we weren't at the levels we needed over the two games and they deserved to go through," said Postecoglou.
"It's disappointing for us because we certainly wanted to make progress in this competition but unfortunately our performances haven't warranted that."
Marseille eased past Qarabag 3-0 in Baku to complete a 6-1 aggregate win in a match that will be remembered for a sporting moment from the hosts telling the referee to rule out a goal they had scored.
Trailing 1-0 on the night, Ibrahima Wadji turned the ball into the net with his hand, but the goal was initially given.
Unlike the Champions League and Europa League, VAR is not in use for the Conference League.
While Marseille players protested furiously, mimicking Wadji's volleyball strike, the referee showed three of them yellow cards, Qarabag coach Gurban Gurbanov spoke to his player.
Wadji then went to Polish referee Bartosz Frankowski, who changed his ruling, after a 10-minute delay.
PSV Eindhoven squeezed past Maccabi Tel Aviv 2-1 on aggregate after a 1-1 draw in Israel, while Partizan Belgrade, Slavia Prague and Vitesse Arnhem all progressed.