Nerves inside the Liverpool dressing room would have been at fever pitch at half-time with the scores tied at 0-0.
Roberto Firmino then doubled the advantage when his shot deflected off Kean Bryan and over a stranded Ramsdale for an own goal.
With four teams above them dropping points, the result is massive for Liverpool's top-four chances.
They also lead Everton by three points, alhough their Merseyside rivals have two games in hand.
Speaking after the game, Jurgen Klopp was visibly delighted with the result.
"The plan was to keep going and do the same stuff again," the German said. "When we could play like we wanted to play, they couldn't deal with us.
"Sheffield United have lost a lot but all by the smallest margin. It's clear these games stay exciting until the very end. We could have scored three in the first half. What we created was very good. The finishing was good enough for today, but we can improve.
"It's normal that we're criticized, people have written us off. That's fine.
"This was about us showing we're still there. We play Chelsea on Thursday, we have to show it again. We have to win football games, we know that. There's no way into the Champions League without results. Tonight we got it, we keep going."