Top seed Edwards, one of the favourites to win a medal in the super heavyweight competition, tried to use his range and keep Lenzi at bay with his jab, but the Italian connected with a couple of powerful right hands in the first two rounds.
The American, who was given a point deduction in the second round, just about edged the third, but it was too little too late.
Australia, who have the largest boxing contingent at the Games with 12 boxers, got off to a difficult start on Monday.
Garside, a Tokyo Olympics bronze medallist, fell short against Richard Kovacs after the Hungarian took charge of the lightweight bout in the second round and showed great flair to down the Australian by unanimous decision.
Kovacs next faces either Rio Olympics silver medallist Sofiane Oumiha of France or Jordan’s Obada Alkasbeh.
“He made the adaptations that he needed to in that second round. All credit to him,” Garside said.
“I just wasn’t able to make the adjustments quick enough. I really envisioned myself winning that gold medal for Australia.”
In the women’s lightweight category, McDonald was also outclassed by her Ecuadorian opponent Maria Jose Palacios, who clinched a unanimous points decision.
The taller Palacios found it easy to break through McDonald’s guard and was comfortably ahead on the judges’ scoresheets.
In Monday’s opening bout, Asian Games champion Yang Wenlu of China comfortably beat Vietnam’s Ha Thi Linh.