He had a special praise, too, for the 39-year-old Buffon on the goalkeeper's 100th Champions League appearance for the club.
Buffon made two early saves from Monaco striker Kylian Mbappe with the score 0-0, thwarted top scorer Radamel Falcao with a fine stop just after the break and denied Monaco a lifeline with a stunning finger-tip save over the bar from substitute Valere Germain's close-range header in the last minute.
"He's still No. 1 in the world," Allegri said. "He is always decisive in the big games."
Little wonder centre-half Giorgio Chiellini rushed over to hug Buffon at the final whistle. "Gigi was perfect," Chiellini said.
Juventus gave a masterclass in how to withstand pressure and then punish on the counter attack, with both of Higuain's goals coming after good Monaco spells.
"This could become an extraordinary season for us," Allegri said of his team's bid for a title treble.
Higuain pounced in the 29th minute, side-footing home after Dani Alves had latched onto his pass down the right before back-heeling the ball back to the Argentina striker as he sprinted into the area.
The Juventus bench rose as one after Higuain slid in to turn home a deep cross in the 59th, with Alves again the provider.
The winner plays either Real Madrid or city rival Atletico, with Real heading into the return leg at Atletico with a 3-0 cushion thanks to Cristiano Ronaldo's hat-trick on Tuesday.
Monaco's vaunted attack was blunted by a solid Juventus defence that has now conceded only two goals in 11 Champions League games.
Monaco attacked relentlessly after the break, with Buffon diving at the feet of Mbappe minutes after his save from Falcao.
"In every game I want to show that I deserve to play at this level, despite my age," Buffon said. "The day I quit, I want people to be sad about it."
Monaco manager Leonardo Jardim praised Buffon.
"With two or three chances they scored twice, and from the same number we couldn't," he said. "But that is also down to Buffon. He made two or three incredible saves."
