The Senegal international caught Azpilicueta with his elbow in the first minute at Stamford Bridge and was booked.
Mane’s yellow card was the fastest in the Premier League since the start of the 2006-07 campaign, committing the foul after just 15 seconds.
Referee Anthony Taylor’s decision proved to be decisive, as Mane went on to score the opening goal of the game just moments later.
Mohamed Salah doubled Liverpool’s lead, but two goals from Mateo Kovacic and Christian Pulisic made it 2-2 at the interval.
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Despite chances continuing to flow, neither side could find a winner. The result leaves Manchester City 10 points clear at the top of the table.
“It’s a clear red,” insisted Azpilicueta when asked by Sky Sports at full-time. “I don’t mind if it’s five seconds into the game or the first action, it’s a clear red.
“He doesn’t want to challenge, he doesn’t see the ball, he just wants to hit with the elbow. I honestly don’t understand.
“We have had these kind of decisions the other day. Two penalties, VAR at the end didn’t come.
“Today, the first action is a clear red card and, of course, we are getting decisions against us. Of course, it could change a lot the way of the game.”
On Mane’s challenge, Azpilicueta added: “I can understand we are promoting the challenges in the Premier League, 50-50 with the ball.
“Sometimes we are seeing a joke of penalties honestly and sometimes we see real dangerous actions and we don’t take action so a lot of meetings we do every season.
“We see that there are a few gameweeks where the referees become a bit softer, soft penalties, so in terms of red cards then it’s opposite because they speak, I don’t know what happens.
“We don’t see the consistency even in the same game or through the season. It’s disappointing these kind of decisions, they’re clear.
“I watched the replay but I didn’t need this because in life I knew this was a clear red.”
Ex-Chelsea striker Jimmy Floyd Hasselbaink thinks Mane was lucky to avoid a red card. “I think it’s reckless,” he said at half-time.
“I think it’s a red card, for me. He knows exactly what he’s doing. It’s an elbow. If you give a yellow card, it has to be a red.”
Chelsea manager Thomas Tuchel was clearly furious with Taylor’s reluctance to dismiss Mane for his controversial challenge.
Sky Sports pundit Gary Neville believes the Liverpool ace avoided a red card because the foul happened very early in the match.
“The Chelsea manager doesn’t think he should be on the pitch,” said Neville on commentary. “It’s not a great one from Mane.
“It’s six seconds into the game and maybe that’s what helps him. I don’t think he looks at Azpilicueta and rocks his elbow back as if he really meant it.”