United went ahead four minutes before half-time when South Korean midfielder Park twisted to get his head on a Nani cross from the right which had taken a slight deflection off Arsenal defender Gael Clichy.
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They squandered the chance to go 2-0 up when Wayne Rooney fired a penalty well over the bar with just over a quarter of an hour left after Clichy had handled in the area at Old Trafford.
The hosts, unbeaten in the league this season and with the best home record, were facing the team with the best away form but Arsenal came up short and they now sit second on 32 points, two behind United who have a game in hand.
United boss Sir Alex Ferguson's tactic of playing a five-man midfield paid off as Arsenal's usually potent combination of Marouane Chamakh, Samir Nasri and Andrei Arshavin could not string strong together the moves that have won much admiration this season.
"We didn't create enough chances. Our passing was not quick and sharp enough to get them out of position," Wenger told a news conference.
"The technical quality of the game was average on both sides because the pitch is very poor in my opinion and the game suffered a lot for it."
CHILEAN MINERS
Watched by some of the Chilean miners who were rescued in October after being trapped for more than two months underground, a patchy encounter will have tested the patience of even those miners.
It also did little to endorse either side's title credentials as they struggled to keep possession when they got near the box.
United were keen to test young Arsenal goalkeeper Wojciech Szczesny who was making his first Premier League and standing in for Polish compatriot Lukasz Fabianski.
Portugal winger Nani shot straight into the 20-year-old's arms and Rooney did the same a couple of minutes later but the keeper stood strong, although his distribution was sometimes poor.
Ferguson had remarked before the game that Arsenal had become a more physical side and his team got increasingly frustrated as a series of clumsy challenges failed to tempt referee Howard Webb to delve into his pocket for a card.
The first card came just before the break, with Chamakh punished for bringing down Nani and then three more followed for Arsenal - with none for United - as the tackles got higher.
United defender Rio Ferdinand admitted it had not been a pretty match.
"These games sometimes you don't play well but you just want to go out and do as well as you can," he said. "Three points is three points."
Arsenal fans started chanting; "We want Cesc Fabregas" at the start of the second half and with less than half an hour to go manager Arsene Wenger heeded their pleas.
Arsenal skipper Fabregas, out with a hamstring injury since November 23, came on with Robin van Persie in a double substitution for the ineffectual Tomas Rosicky and Jack Wilshire.
But it was United who created the better chances and Arsenal had Szczesny to thank for some excellent saves, first from Anderson and an attempted lob by Rooney.
Rooney had sent his spot-kick high into the stands and Arsenal may feel that deserved to happen because Clichy had been somewhat unlucky to give away the penalty as he could not move out of the way when Nani was dribbling past him.
The hosts held on, while the biggest cheers of the night were reserved for the rescued miners who posed for photos in the stands.
