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.
Spurs are joined in the group phase by Ajax, Auxerre, FC Copenhagen and MSK Zilina.
Uruguay World Cup striker Suarez got the first goal as four-times European champions Ajax overcame Dynamo 3-2 on aggregate to reach the group stage for the first time since 2005/06 - the last occasion their Ukrainian opponents failed to qualify.
Slovakian champions Zilina reached the group stage for the first time, beating Sparta Prague 1-0 thanks to a spectacular bicycle-kick from Gambian Momodou Ceesay to complete a 3-0 aggregate win.
FC Copenhagen went through on away goals, beating Rosenborg Trondheim 1-0 for a 2-2 aggregate draw, and Auxerre beat nine-man Zenit St Petersburg 2-0 to qualify for the first time since 2002-03 with a 2-1 aggregate win.
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 Defoe incident then killed off Young Boys as the England striker clearly used his upper arm before turning to score with a low shot off the post.
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.
Crouch headed another from a corner just past the hour mark and completed his hat-trick with a 78th minute penalty.
"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.
Zenit St Petersburg could not match a spirited challenge from Auxerre and had two players sent off in the second half.
The team from Burgundy, who had lost the firs
