Curtis Jones’ first-half brace and Trent Alexander-Arnold’s stunning free-kick secured a seventh successive Premier League win.
It leaves Klopp’s side a point behind United and Newcastle, having played one more game.
At full-time, the German boss heralded the vocal away support with his routine chest pumps as Liverpool hold onto hopes of sneaking into the top four.
And Gary Neville believes that the two clubs that Liverpool are chasing will have viewed the gesture as a warning that their Champions League statuses aren’t safe.
He told Monday Night Football: “They [Newcastle and United] have got a job to do.”
“And Liverpool, Jurgen Klopp’s little fist pump at the end, is definitely something if you’re a Newcastle or Manchester United player that you recognise he’s coming for you.
“I think he’s sending them a message, and they’re going to have to make sure they do it properly because Liverpool will win their two games.
Manchester United and Newcastle have got to forget the idea that Liverpool might drop points, they’ve got to do their jobs, or else they’ll end up out of the top four.”
Liverpool host Europe hopefuls Aston Villa at Anfield on Saturday before travelling to already-relegated Southampton on the final day.
Newcastle and United know they can secure Champions League football next season by winning two of their final three games.
The Magpies face Brighton and Hove Albion and Leicester at St James’ Park before a visit to Chelsea.
Meanwhile, Erik ten Hag’s side will attempt to shake off their torrid away form at AFC Bournemouth on Saturday before a home double-header against Chelsea and Fulham.
Liverpool guaranteed at least Europa League football with their latest triumph, but Klopp and his squad will be eyeing much more.
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