The Chilean was sent off for a second bookable offence just one minute after Lautaro Martinez had fired in a superb opener to make the tie 2-1 to the Reds on aggregate.
Any hopes of the Italian side finding an equaliser, though, were dashed by the decision to dismiss ex-Arsenal and Manchester United star Sanchez, despite the second foul being debatable.
His first yellow card was straightforward, catching Thiago's right knee with his studs but the second was a lot less clear.
After sliding to the pitch to nudge the ball clear of Fabinho, his momentum took him into the Brazilian's leg, despite attempting to retract his own outstretched limb.
Referee Antonio Mateu, from Spain, felt it was enough for a second caution, leading to Sanchez's dismissal but Ally McCoist wasn't so sure - even if Laura Woods did feel the call was correct.
Queried about it on the talkSPORT Breakfast, the Scottish striker said: "Poor. No, I'm not having it. I'm absolutely not having it. He wins the ball clearly. He follows through and, if anything, the players meet.
"The game is gone. It has. It's gone. It's never a second yellow. Never."
Woods attempted to talk McCoist round and replied: "I have to say, when I saw it, I thought because it was a second yellow he should go."
And after her co-host and sometime sparring partner joked she was 'obviously wrong', she added: "Maybe I am. I wouldn't dream of arguing with Ally McCoist."
He wasn't finished either, adding: "I just think there is no common sense. The way he goes into the tackle, there is no threat. There is no danger. It's just a natural follow through, for me. I just think some common sense should be shown. I really do."
"He's won the ball. Are you telling me that he shouldn't go to win that ball? Look how low his leg is. He's bringing his leg away. In slow motion it makes it look worse.
"First and foremost he's stupid for the giving the referee and opportunity to make the decision. No one can debate that but I still think it's harsh to give a yellow card.
"It's a foul, yes, in today's game but I don't believe it's a yellow card."
One man went a lot further on the call and believes it's Fabinho who should be in the dock.
On beIN Sports, former Arsenal manager Arsene Wenger, who signed Sanchez for the Gunners, said: "(It's) borderline between cheating and being clever.
"Fabinho on this foul was on the other side… he was cheating. He made more of it."
Meanwhile, over on American coverage of the match, a host of Premier League heroes also agreed the red card was incorrect.
Arsenal legend Thierry Henry said: "I'm sorry if you've played football and I think sometimes the referee just goes by the book.
"What are you supposed to do? He gets the ball, your foot is going to go up because you're kicking the ball away.
"We see on the replay full speed you don't have time to take the ball away by kicking the ball. If he (the referee) plays football that's not a yellow."
Manchester United icon Peter Schmeichel explained Henry's point further, addding: "You go forward with your leg and at speed, there's no way you can pull the leg out. Sanchez wins the ball that's enough."
And Liverpol legend Jamie Carragher added: "Sometimes a foul or situation like that where I probably have less sympathy was if Alexis won the ball with his studs.
"He doesn't win the ball with his studs, he wins it with his foot which has to go somewhere.
"It's not even that high it's on the ankle or the boot. It's probably just a foul and that's debateable.
"The fact he went into his boot it's nowhere near a yellow card but you could argue the two fouls put together is a red card maybe."