Steffen took an extra touch after receiving a back pass from defender John Stones, giving Mane enough time to slide in and poke the ball into the net to grab Liverpool's second goal of the match.
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"Today is special -- we were playing one of the best teams in the world," Mane said. "To win this kind of game, especially in a semifinal, is a big, big, big win.
"We started very well, everybody started on the front foot -- for my first goal the goalkeeper made a mistake but I think we pushed him to make that mistake.
"That is our style and I think that made the difference."
Liverpool opened the scoring on the ninth minute when Konate headed in ball from a beautifully weighted Andy Robertson corner.
Their lead doubled just 10 minutes later when Mane chased down Stones' pass to Steffen, and capitalised on the goalkeeper's casual touch of the ball.
Just before half-time, Mane found his second after picking up a clever pass from Thiago Alcantara and volleying it past Steffen.
After the break, City replied quickly with Jack Grealish pouncing on a square ball from Gabriel Jesus and firing it into the net on the 46th minute.
The goal gave City momentum, and the side went close two more times through Jesus. However, the efforts were well-covered by Liverpool goalkeeper Alisson to maintain the two-goal lead.
With 90 minutes on the clock, Bernardo Silva added a second for City, but despite a nervy four minutes of extra time, Liverpool were able to hold on and secure the victory.
"Absolutely proud, incredible -- I think the first half was one of the best we ever played," Liverpool manager Jurgen Klopp said.
"We did all the right stuff, we scored in the right moments, we played an incredible game in the first half, I really loved each second of it," he added.
For City it was the third straight season in which they have lost in the last four of the FA Cup.
"We gave everything after a tough first half," said City boss Pep Guardiola.
"For the first goal, the set pieces they are very, very strong. The second was an accident. Difficult to come back but they did it perfectly in the second half," he added.
Liverpool remain in contention for four trophies having won the League Cup, reached the semifinals of the Champions League and still being in the race for the Premier League title where they sit a point behind leaders City.
Before Saturday's match, the period of silence in honour of the Hillsborough victims was interrupted by fans chanting and City apologised to Liverpool following the incident.