Inter trailed 2-0 heading into the Champions League last-16 second leg and was fortunate not to be further behind as Joel Matip and Mohamed Salah hit the frame of the goal.
However, Lautaro Martinez's superb 20-yard strike with just over an hour played changed the mood around Anfield and gave Inter serious hope of salvaging extra time.
Inter's prospects of staying in the competition were badly dented 107 seconds later, though, as Sanchez received a second yellow card for following through on Fabinho.
Former Arsenal and Manchester United forward Sanchez had been yellow carded late in the first half for a similar lunge on Thiago Alcantara.
Inter, which conceded late on to Roberto Firmino and Salah in the first leg, offered little attacking threat with 10 men and were lucky not to concede when Salah again hit the post.
Inzaghi is known for substituting players when they are at risk of being dismissed, but he did not feel the need to bring off Sanchez in what was an at times bad-tempered game.
"Absolutely not," he said after his side's 1-0 win on the night.
"In general I am very careful, but at that moment I needed Sanchez on the field. I was not going to change him on an evening like this."
Sanchez is the first Inter player to see red in the knockout stages of the Champions League since Cristian Chivu against Schalke in 2011.
Inter boss Inzaghi did not comment on whether he felt the red card was harsh as he did not see the second foul, though he felt his side deserved more over the two legs.
“We feel upset about the fact we had a player sent off just when we were getting the upper hand in the game," he said of Sanchez's first dismissal in 64 games in the competition.
"There are certainly great regrets for us as we wanted to reach the last 16.
"But we were up against Liverpool, who along with Manchester City and Bayern Munich are the best team in Europe right now, and we fought them on an even keel.
"We are probably paying the price for the final 15 minutes of the first leg, because I think we had two very good games against an excellent opponent.
"I think the first 75 minutes saw us deserve far more, then we conceded on a corner, which wasn't even a good one, and the second goal really knocked us down.
"During our best period of the game and the worst for them, Alexis Sanchez was sent off, and that was bad timing.
"Over the two legs we showed we are equal to a really strong Liverpool team. We played well and I am more bitter about the game at San Siro than what happened in this game."
Martinez's goal was his first in the Champions League in 11 appearances since netting against Real Madrid in November 2020.
It proved to be nothing more than a consolation over the two legs, though it did earn the Italian champions a first win against Liverpool home or away since the 1964-65 season.
But at the end of 180 minutes, Martinez was left to rue Sanchez's red card at a pivotal moment in the game.
"We did what we wanted to by showing character," he said. "It was a pity to then go down to 10 men because these are the details that make the difference.
"All we can do now is look forward. We are out of the Champions League but still have the Coppa Italia and Serie A. We must learn from these mistakes."