De Minaur produced a solid performance to clinch his seventh win in Milan, and distanced Andrey Rublev's previous record of six.
"I knew from playing Casper before that he'd dictate any short ball without a purpose," De Minaur said.
"The game plan was to play on my own terms, play aggressively, play different styles of tennis. The goal I had was [to take] any half-chance, go up and back myself at the net.
"I'm glad it worked out today. I'm looking forward to the semifinals."
Kecmanovic prevailed in 58 minutes by easily winning the first two sets, before battling on a tie-break in the last set to finish 4-1, 4-1, 4-3(6).
De Minaur's semifinal opponent will be second seed Tiafoe, who earned his first trip to the semifinals in Milan by beating the opponent in straight sets 4-2, 4-2, 4-2.
On the last game of the night, Italy's Jannik Sinner, who had already advanced to the semifinals, suffered his first defeat of the tournament, giving up 4-3(5), 3-4(3), 4-2, 4-2 against Frenchman Ugo Humbert.
Sinner, 18, will face Kecmanovic in the semifinals, with De Minaur taking on Tiafoe.
Matches at the ATP NextGen finals are played on a best of five sets, first to four games in each set basis.