The Reds were leading 1-0 thanks to a goal from Sadio Mane after two minutes and 15 seconds at Goodison Park, the fastest scored in this fixture in the Premier League era.
Virgil van Dijk has been flooded with support from his Liverpool team-mates after it was revealed he would undergoing surgery for a serious knee injury suffered in Saturday's Merseyside derby against Everton.
However, he was forced off after 11 minutes at Goodison Park after being on the receiving end of a wild challenge from Jordan Pickford, who escaped punishment because the Dutchman was offside but could yet face a retrospective ban.
However, the Reds dropped more than just points as they lost their influential defender Virgil van Dijk to a awful injury in the game.
The club did not specify a timescale for the 29-year-old defender's return to action.
"Great result and great character to come back, led by a very inspirational captain", Solskjaer told Sky Sports.
The Anfield outfit is out to defend the Premier League title they won for the first time in 30 years last campaign, while they also redeem their image in the Champions League this time around.
Klopp was also not happy with another reckless Everton challenge that earned Richarlison a straight red card for taking out Thiago Alcantara.
Liverpool fans were furious that a seemingly late Jordan Henderson victor was disqualified at Goodison Park when Jurgen Klopp's side drew 2-2.
Michael Keane equalised before Mo Salah's 100th Liverpool goal had them ahead with 15 minutes left.
However, no penalty was awarded after a VAR review because Van Dijk had been marginally offside and Pickford was lucky to avoid a red card.
I think Liverpool certainly have to go into the market in January, and not just because of Van Dijk's injury, but Liverpool were weak in that area anyway.