The 17-year-old became the youngest man in 40 years - since Boris Becker won at Wimbledon - to reach the quarter-finals of an ATP grass-court event but he proved no match for Auger-Aliassime in a 7-6 (7-3) 6-3 defeat.
Engel had claimed his first professional victory on grass earlier this week when he beat Australian James Duckworth, following it up with an upset win over US seventh seed Alex Michelsen, but beating the experienced Canadian proved to be a step too far for the teenager in Friday's quarter-finals.
"His serve was so incredibly good today that I felt pressure in my own service games," Engel said.
"But it was great fun playing in front of this crowd again. I can't wait to play here again next year."
Top seed and home favourite Alexander Zverev beat Brandon Nakashima 7-5 6-4 to advance to the semi-finals while Americans Taylor Fritz and Ben Shelton also reached the last four.
Zverev is seeking his first grasscourt title and was involved in a tight opening set which was tied at 5-5 when the German switched gears in the final two games to clinch it.
But Zverev was far more dominant against his American opponent in the second set as he finished the match with 31 winners in a contest that lasted 90 minutes.
He will play third seed Shelton next after the American, who was virtually unstoppable on his first serve, beat Jiri Lehecka 6-4 6-4, firing down 18 aces.
Shelton was so dominant in the second set that he did not lose a single point on serve while he won 100 per cent of his first-serve points in the match, with victory ensuring he will break into the top 10 for the first time in his career.
Marton Fucsovics had beaten Fritz in straight sets in Stuttgart two years ago but this time the American, who has yet to drop a set at the tournament, came out on top in a dominant 6-3 6-4 victory.
In the other pre-Wimbledon grass-court event going on in the Netherlands, American lucky loser Reilly Opelka put up an intense battle to stun top seed Daniil Medvedev 7-6 (7-5) 7-6 (7-5) and reach the semi-finals in Rosmalen.
Medvedev served up three double faults in the second set tiebreak and five overall, but there were no such errors from Opelka, who slammed down 24 aces.
The giant Opelka will contest a grass-court semi-final for the second time in his career and faces Estonian qualifier Mark Lajal or Belgium's Zizou Bergs on Saturday for a spot in the final.
French second seed Ugo Humbert will play Canadian Gabriel Diallo in the other semi.
Diallo knocked out third seed Karen Khachanov 7-6 (8-6) 6-4 while the Frenchman eliminated Portuguese Nuno Borges, seeded seventh, 6-1 6-4.
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