A Romelu Lukaku brace inspired Everton to a 3-1 second leg victory at Goodison Park.
Lukaku cancelled out Sekou Sanogo's 13th-minute opener before fellow Belgium international Kevin Mirallas sealed a 7-2 aggregate success against the Swiss club.
Everton are the only Premier League team remaining in the competition after Liverpool and Tottenham bowed out on Thursday, something that pleases Martinez.
"It was an interesting game to see how we could cope with playing in a second leg," he explained. "We had such a good lead and sometimes that can affect the way you want to perform.
"We knew Young Boys had a fantastic result against FC Basel in their domestic league and I admire their team because they are an attacking side who want to get on the ball.
"We made it an open game. I was very satisfied with the way we scored three goals and created a number of good opportunities.
"To be the last British team left in the Europa League is a very satisfying achievement and we have been very, very pleased with the way we have played both games in this particular tie."
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