Mike Williamson was the fall guy for Newcastle United as his 76th-minute own goal meant Championship Brighton and Hove Albion knocked out the Premier League side 1-0 in the late kick-off.
Fourth-tier Crawley Town stunned second-division Hull City by the same score in the only other shock, while Chelsea also beat Queens Park Rangers 1-0.
United left-back Evra was given a hostile Anfield reception after the Frenchman's involvement in a spat with Liverpool's Luis Suarez in October, which led to the Uruguayan forward's current eight-match ban for racial abuse.
Kenny Dalglish's side, still missing the verve offered by Suarez, took the lead on 21 minutes when Daniel Agger headed in a corner that struggling United goalkeeper David de Gea failed to collect after he was crowded out by the hosts.
Asked if Liverpool had merited victory, captain Steven Gerrard told ITV Sport: "Not on possession, probably not on general play; we've got to give Manchester United credit, they controlled large periods of the game today, but the most important thing was the result and we got there in the end."
"I don't know how we lost it, it was a really good performance by us," he said.
Dutchman Kuyt's strike sent Anfield into raptures and prompted more hopes in the city that Dalglish can bring back the glory days from his previous spell in charge.
Racism accusations were also the backdrop to Chelsea's win at Queens Park Rangers, where John Terry, in court next week after being charged over comments allegedly made to QPR's Anton Ferdinand in October, was kept away from his fellow defender as the clubs decided not to shake hands before kick-off.
