Roberto Martinez???s side were given an early scare at Goodison Park after Sekou Sanogo tapped home from close range to give the visitors an early lead.
Minutes later Scott Sutter hit the bar with a header for the Swiss side but any worries Everton may have had disappeared as Romelu Lukaku hit back from the spot in the 25th minute and then fired the Toffees into the lead with an excellent volley just five minutes later.
The turnaround was complete when Kevin Mirallas struck just before the break on what, in the end, was a comfortable night for Martinez???s men.
Everton will now have all eyes firmly fixed on tomorrow lunchtime???s Europa League last 16 draw as they are left to fly the flag for Britain in this season???s competition.
Young Boys may have had it all to do at the start of the night after last week???s 4-1 defeat in the first leg but they started positively with the English-born Sutter and Yuya Kubo both having early attempts.
Despite their three-goal advantage, Everton looked anxious and their nerves were compounded when Sanogo tapped home from close range in the 13th minute after Tim Howard could only parry a shot from the edge of the area.
The visitors had a glimmer of hope and it could have got even better minutes later when Sutter???s header came crashing back off the cross bar.
Everton were in disarray at the back, however, they were back on level terms in the 26th minute when Steven Naismith was brought down in the box by goalkeeper Marco Wolfli. The referee had no hesitation in pointing to the spot and Lukaku fired the ball into the back of the net for his 14th goal of the season.
Any nerves the Goodison Park faithful had were soon gone when full-back Luke Garbutt streaked clear down the left and picked out Lukaku in the box, who connected with a sweet volley to give the hosts the lead.
The turnaround was complete just before the break when Darren Gibson released Mirallas through on goal and the Belgium international calmly slotted the ball past the onrushing Wolfli.
With the game seemingly wrapped up Martinez took Lukaku out of the firing line but it would have been disturbing for all to see the Belgian striker clutching his hamstring as he left the field.
The change did not halt Everton as substitute Arouna Kone led the home charge before finding Mirallas, but his shot was well saved low down by Wolfli.
With a trip to Arsenal on the horizon, Martinez started to ring the changes, replacing James McCarthy with Muhamed Besic.
Everton, who were in full control at this stage, still pressed forward and this time it was Mirallas picking out Kone but again his shot was well saved.
Kone was in the thick of the action again, hitting the post with 15 minutes to go before Leon Osman had a goal ruled out with just minutes left on the clock , but it was not to matter as Everton turn their attention to the last 16 draw in Nyon tomorrow lunchtime.