The Bianconeri coach saw his side lose for the first time since his arrival despite Cristiano Ronaldo's goal putting the champions in front at the Stadio Olimpico.
Luiz Felipe headed Lazio's equaliser before the turning point of the match, Cuadrado seeing red after bringing down Manuel Lazzari as he ran through on goal.
Referee Michael Fabbri initially opted to book Cuadrado but then changed the yellow card to red after consulting pitchside monitors, and Lazio made the most of their man advantage.
Sergej Milinkovic-Savic brilliantly scored from Luis Alberto's lofted pass, and Felipe Caicedo struck deep in added time to secure Lazio's first home league win over Juve since 2003.
"We didn't respect our duties, because players were in the wrong position," Sarri told DAZN of the incident leading to Cuadrado's dismissal. "It was a distraction and you pay for that.
"We had decided before the game on certain positions and we didn't cover them.
"I do not entirely agree that it was a clear scoring opportunity, as the ball was going wide. I thought a yellow card was the right ruling, but the fact remains that is out of our control and we made a mistake to create the situation."
Sarri, who said Rodrigo Bentancur needs tests on a ligament injury, felt there were positives to take from Juve's performance despite a second straight league game without a win.
"In the second half, when the game seemed to be in a more balanced state, the red card changed everything," he said. "We lost the match, but, to be honest, I saw improvements in my team."
Lazio have won seven games in a row in Serie A and are only five points behind leaders Inter, prompting suggestions they could be in the title race.
They last won the league 20 years ago, and coach Simone Inzaghi feels topping the table may be a realistic target for his impressive side.
"We are in a great period of form," he told DAZN. "We should be ambitious, knowing also that six weeks ago people said we weren't getting results but I remained calm.
"The club was always close to me, even when we lost, reassuring me there were no problems.
"I also have a squad that is always at the disposal of my staff and I. Now we enjoy this victory, which we hadn't managed for 16 years here, a game I remember, because I was on the bench.
"We enjoy it for a few hours, but from Thursday we must get right back on track and aim to stay in the Europa League. We played well, created many chances and could have scored more goals, so I praise these lads for something extraordinary."