Lukaku scored a hat-trick in the 4-1 first-leg victory at the Stade de Suisse, and added two more at Goodison Park to fend off an early fightback from the Swiss side and seal progression.
Young Boys reduced the three-goal deficit with Sekou Sanogo's 13th-minute opener but a two-goal salvo from Romelu Lukaku inside half an hour took any glimmer of hope away from the Swiss side.
Kevin Mirallas added a third before the interval - before Lukaku was given a breather early in the second half ahead of Sunday's Premier League clash with Arsenal.
Even without Lukaku to deal with, Uli Forte's side were unable to make any headway as they are left to focus on their Swiss Super League title challenge after a convincing defeat over the two legs.
Roberto Martinez and Everton will look forward to their first European last-16 tie since 2008, having progressed shortly after rivals Liverpool were sent out of the same competition on penalties against Besiktas.
Despite trailing by three goals in the tie, Young Boys began well against an Everton side showing five changes from Sunday's 2-2 draw with Leicester City.
Forte's men came into the game after a 4-2 win over Swiss champions Basel and were given hope when Sanogo prodded home from close range amid questionable goalkeeping from Tim Howard and Everton's offside appeals.
After Rapha Nuzzolo fired an effort at goal, Milan Vilotic got a touch with his head and, when Howard fumbled, Sanogo was on hand to nudge in the opener.
Scott Sutter - one of five to come in from the win against Basel - crashed Gonzalo Zarate's corner off the crossbar but the visitors' momentum was curtailed when Marco Wolfli upended Steven Naismith in the area.
Lukaku's backheel caused the initial danger and the Belgium international took his Europa League tally to five from the spot before volleying home Luke Garbutt's cross from the left five minutes later.
When Mirallas slotted Darron Gibson's pass beyond Wolfli, the game was put further beyond Young Boys' reach with Lukaku replaced by Arouna Kone shortly after the break.
With the tie all but over, and Everton's focal point Lukaku given the remainder of the game off, chances dried up as Martinez introduced Muhamed Besic for James McCarthy in midfield,
For all Young Boys' tidiness on the ball, they offered a limited attacking threat, while Everton came closest to a fourth when Kone struck the post late on before Leon Osman's header was disallowed due to a foul from the substitute on Vilotic.
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