There were some tasty challenges from Everton in the early minutes - Ross Barkley continuing his trend of disciplinary invincibility in these fixtures - but Anfield erupted in the 8th minute when Liverpool composed themselves and Mane powered through the opposition half to fire in the opener. After the Merseyside derby Everton will take on Manchester United live on BT Sport (Tuesday 4 April).
Coutinho has struggled since recovering from an ankle injury in January but provided an assist as well as his goal as the Reds extended their Premier League run against their Merseyside neighbours to 14 games.
"Yeah [Klopp and I disagreed] because I don't like coaches from the bench the whole time shouting to referees and linesmen, making a big show. Why you need to be afraid, I don't understand that", Koeman told reporters on Friday. They were insane and they didn't need an official on the pitch.
"I saw after we won the last two games the Evertonians were still around", said Klopp. "It's football - a hard and fair game", he said to Sky Sports TV.
Jurgen Klopp's men took an eighth minute lead through Sadio Mane's 13th Premier League goal of the season, only for Ronald Koeman's side to hit back through Matthew Pennington after 28 minutes.
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Liverpool seemed content with their lead so Everton were given a touch more freedom but teenager Dominic Calvert-Lewin had their last chance and should have done better with his close-range header after Simon Mignolet did well to parry Mason Holgate's deflected shot. Because parking the bus and counterattacking is the way to beat Liverpool and Everton for some reason played into Liverpool's hands with their strategy.
Liverpool lost Mane to injury after the break, although that did not damage Jurgen Klopp's men, Divock Origi striking within four minutes of coming on to replace him to seal a crucial victory and star man Coutinho providing the assist.
"When I saw them I thought they would actually look exhausted, but they weren't", he continued. The Toffees were miserable in stretches and the mentality change he was calling for in Derby games starts with him, and we didn't see it today. Having one day training to bring 10 different countries where the players come from after global duty, 10 different ideas, 10 different jobs to do and you have exactly 35 minutes - the same for all of us - to bring it to kind of one idea again.
"If you saw two harsh challenges then say it, if not don't use me to say it".