Monaco, last season's quarter-finalists, moved into the playoff round after the decisive second-leg win over the Swiss side.
The Ligue 1 team — who reached the 2004 final where they were beaten by Portuguese giants Porto — were already 3-1 up from last week's first leg in Bern and finished the job handsomely at the Stade Louis II thanks to a blistering second-half display.
"I liked our performance. We managed to put our game plan into practice — to contain the opposition at first, then counter-attack after we scored the first goal," said Monaco boss Leonardo Jardim.
"We took control of the game,” Jardim said.
“The first half was very good from a strategic perspective,” he said.
“Our goal was to not concede before half-time and force our opponent to attack. That's exactly what happened."
Young Boys, who were semifinalists of the European Cup back in 1958/59, retained some hope of getting back into the tie with the second leg goalless at half-time.
But Monaco — third in France's Ligue 1 last season — struck four times in 23 devastating second-half minutes to inflict a crushing 7-1 aggregate defeat on the club from the Swiss capital.
Ivan Cavaleiro — a big-money summer arrival from Portuguese champions Benfica — curled in a superb opener after 54 minutes before further goals from Layvin Kurzawa, Anthony Martial and another new signing Stephan El Shaarawy saw Monaco run riot.
Elsewhere, four-time European champions Ajax were eliminated from the Champions League round after a humiliating 3-2 second-leg defeat to Rapid Vienna.
Croatian champions Dinamo Zagreb survived a thrilling comeback by Norwegian side Molde to progress on away goals after a 3-3 second-leg draw.
With the tie level after a 1-1 draw in Zagreb last week, the visitors looked to be out of sight after scoring three goals in the opening 22 minutes at the Molde Stadion.
The Norwegian champions — who missed two penalties and had Vegard Forren sent off — hit back with three second-half goals to leave Zagreb hanging on, but the hosts couldn't get the fourth goal that would have put them through at the visitors' expense.
Cypriot champions APOEL remain on course for a second successive appearance in the group stages despite a 1-0 second-leg loss to Danish champions FC Midtjylland in scorching heat in Nicosia.
In 35C temperatures, the Cypriots — who played for an hour with ten men after Vinicius' straight red card — were beaten by Erik Sviatchenko's early goal but progressed on away goals after a 2-1 first-leg success last week.