Trailing 3-2 following last week's first leg in Switzerland, Harry Redknapp's men edged ahead in the tie courtesy of Crouch's early header, and appeared in cruise control midway through the half with Jermain Defoe firing home following the use of an arm.
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Crouch completed his treble after the break, first heading home Gareth Bales's corner, before slotting home from the penalty spot after the in-form Welsh wideman had been upended.
Tottenham, who last appeared in Europe's top club competition in 1962, were ahead in the tie for the first time by the fifth minute as Crouch was left unmarked to head home.
The officials' failure to spot the offence will raise questions over the effectiveness of the two extra linesmen behind the goals who are being used in the Champions League this season.
"I always felt it was possible to get Spurs into the Champions League," said Spurs manager Harry Redknapp. "The club had underachieved for a long time.
"It was important we didn't waste the opportunity. We nearly did in Switzerland but we finished the job tonight."
Young Boys coach Vladimir Petkovic said: "It was difficult after the first goal and the second for me was the decisive moment."
In Amsterdam, Uruguay forward Suarez put the Dutch team ahead in the 43rd minute, latching on to Jan Vertonghen's deflected free-kick and rounding Maxym Koval to score from a narrow angle.
Morocco's Mounir El Hamdaoui scored the second for Martin Jol's team in the 75th minute but there was a tense finish after former AC Milan and Chelsea striker Andriy Shevchenko pulled one back with a penalty.
