Mancini's Manchester City did their job by winning 3-0 at home to Stoke City but the Premier League champions remain seven points behind rivals United following Van Persie's masterclass, with Tottenham Hotspur up to third after a 3-1 victory over Reading.
Dutchman Van Persie was pinpointed by Mancini in the pre-match buildup as the reason why United may snatch back their title and the former Arsenal hotshot continued 2013 where he left off in 2012, curling in a beauty just before the break.
A late tap-in took Van Persie's total to 16 league goals this season as the leaders powered on to 52 points from 21 games, their best tally at this stage in six years.
Arch poacher Javier Hernandez, who had one effort ruled out for offside, gave United the lead at lowly north west rivals Wigan when he slotted the ball in after Patrice Evra's effort was saved in the 35th minute.
The Mexican then grabbed his second after 63 minutes to underline his abilities as an able deputy to the injured Wayne Rooney.
"Wigan are a difficult team to play against, particularly at their own ground, so 4-0 is a good result for us," United manager Alex Ferguson told the BBC.
"The game started slowly but once we got the first goal we were away and we were happy to go in at halftime 2-0 up," added Ferguson.
"Javier Hernandez is a goal-scorer, a real penalty box player. Van Persie's ability is exceptional and his first goal was magnificent - we expected him to be a major player and add something to us and that is exactly what he has done."
The New Year brought a rare goal for City defender Pablo Zabaleta before in-form Edin Dzeko doubled the lead after the break to show Mancini that his strikers are still top class despite the failure to sign Van Persie from Arsenal in August.
Sergio Aguero, who later limped off, squeezed in a penalty to make it 3-0 after Steven N'zonzi had clipped the heels of David Silva - possibly outside the area.
"Today we did a good performance," Mancini told Sky Sports.
An usually fragile Stoke were completely outgunned by City for most of the first half, much as Aston Villa were at Swansea City as their run of three defeats while conceding 15 goals and scoring none looked set to continue before they sealed a 2-2 draw.
Wayne Routledge rounded the goalkeeper to give the Welsh side a deserved advantage but Paul Lambert's young Villa team summoned some fighting spirit after a difficult few weeks and Austrian Andreas Weimann netted just before the interval.
Christian Benteke then coolly converted an 84th-minute penalty after Nathan Dyer's foul but 16th-placed Villa's victory hopes were denied by a stoppage-time leveller from Danny Graham.
Fifth-placed Arsenal were brought back down to earth after Saturday's thumping 7-3 win over Newcastle United when they were held to a 1-1 draw at struggling Southampton and only scored through a Guilherme do Prado own goal.
His 41st-minute equaliser, after Gaston Ramirez's opener arrived after some woeful Arsenal defending, came from Theo Walcott's cross but otherwise the former Southampton man flattered to deceive following Saturday's hat-trick.
North London rivals Spurs moved back up to third, pending Chelsea's home game against bottom club Queens Park Rangers on Wednesday (1945 GMT), despite going behind early to second bottom Reading at White Hart Lane.
Pavel Pogrebnyak netted after four minutes after reacting quickest when Ian Harte's free kick came back off the woodwork.
Michael Dawson quickly equalised with a header from a corner before Emmanuel Adebayor leapt to head his first league goal of the season and Clint Dempsey added a late deflected third.
Fulham winger Alex Kacaniklic secured a 2-1 win at high-flying West Bromwich Albion to end a sorry run of results for the Londoners who had gone in front through Dimitar Berbatov after 39 minutes.
West Brom improved markedly after the break and on-loan Belgium forward Romelu Lukaku is likely to see increased media speculation about a possible recall by parent club Chelsea this month after capping a superb display with a 49th-minute tap-in.
The hosts then hit the post twice but 13th-placed Fulham would not be denied and Swedish winger Kacaniklic expertly spun on the edge of the area before stroking the ball home 13 minutes into the second half.
Mid-table West Ham United eased to a 2-1 home win over stuttering Norwich City.
As well as Chelsea's fixture on Wednesday, Liverpool welcome Sunderland (1945) and Everton head to Newcastle United (2000).
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