Van Persie netted from the spot nine minutes from time to secure Manchester United's first Anfield win since December 2007 against a Reds side, who were a man down for most of the second half following the dismissal of Jonjo Shelvey.
The hosts still managed to take the lead through Steven Gerrard's volley seconds after the interval, but Rafael equalised soon after and Van Persie's late spot-kick proved decisive on an afternoon which earlier saw both clubs pay tribute to the 96 victims of the Hillsborough disaster following the publication of the independent panel's report which exonerated fans.
Elsewhere, Laurent Koscielny's late equaliser earned Arsenal a deserved point at the Etihad Stadium.
The Gunners had much the better of the first half but could not make their dominance count and Joleon Lescott headed the champions in front from a 40th-minute corner.
That looked like being the way it would stay as Arsenal struggled to find a way to goal, but Koscielny proved the unlikely hero with an unstoppable shot eight minutes from time.
Jermain Defoe scored his sixth goal of the season as Tottenham came from behind to beat QPR 2-1 - their first home win under Andre Villas-Boas.
A dreadful first-half performance meant Villas-Boas left the pitch at half-time to a chorus of loud boos, but his team turned things around after the break, with Defoe's strike and an own goal from Alejandro Faurlin handing the Portuguese his first win at White Hart Lane at the fourth attempt.
Former Spurs striker Bobby Zamora had put Rangers ahead in the first half.
Finally, Demba Ba's fourth goal of the season fired Newcastle to a 1-0 victory over Norwich.
Strike partner Papiss Cisse missed a penalty on the stroke of half-time but it did not prove costly for the Magpies as they claimed their second Premier League win of the season.
Sunday's Premier League Results
Liverpool 1-2 Man United
Newcastle 1-0 Norwich
Man City 1-1 Arsenal
Tottenham 2-1 QPR