The Londoners would have gone top with a win against the champions but few chances were created in a disappointing 0-0 draw at the Emirates.
"Our defensive focus was extreme today because we conceded so many goals on Saturday. It was important not to lose tonight," manager Wenger said.
"United restricted us to very few chances but our team was highly focused not to concede tonight and maybe that restricted our game going forward.
"We were certainly hit by the five goals we conceded on Saturday and you could feel that a little bit tonight. I think we were nervous where we are usually very dangerous breaking through the middle with speedy passing," the Frenchman added.
"We lost two points tonight but the point keeps us in the race."
Arsenal are second, one point behind Chelsea, while United are seventh, 15 points off the pace.
Wenger's side have taken just five points in six games against Chelsea, Manchester City (in third place), Liverpool (fourth) and United.
Arsenal let in four goals in the opening 20 minutes against Liverpool and Mikel Arteta's mistake after 90 seconds almost led to another early setback.
Szczesny, however, saved the home side from another terrible start by guessing the right way as Van Persie shot from close range with his favoured left foot.
Good chances, though, were few and far between in a dislocated and often frantic first half between two sides searching for their best form.
Mesut Ozil, Arsenal's £42 million buy from Real Madrid in August, was more involved than at Anfield at the weekend.
"He was one of our best players," said Wenger.
Van Persie, returning to the club he represented for eight seasons, shot across goal when well placed after 18 minutes but United again failed to make the most of new midfield signing Juan Mata.
Jack Wilshere was at the core of most of Arsenal's best moments and received consistently rough treatment from the visitors.
United full-back Rafael, injured in an aerial challenge with Giroud, was replaced by Rio Ferdinand for the second half which triggered a reshuffle of the back line but the match followed the same pattern of high tempo movement but little penetration and no end product.
Laurent Koscielny had a header from a corner cleared from under the United crossbar by Antonio Valencia after 63 minutes and Giroud narrowly failed to connect with a cross in front of goal.
Wayne Rooney, agitated and buzzing in the first half, all but disappeared in the second period before setting Van Persie up for his late header.
Arsenal were robbed of possession and suddenly Van Persie came to life. He fed Rooney and raced into the area where he met the England forward's cross with a meaty header which the Polish keeper pushed on to the bar and away.
David De Gea was then forced into a diving one-handed save to thwart Santi Cazorla late on and, as the 90 minutes ended, the United goalkeeper plunged to keep out another low effort from the Spaniard.
"We defended well throughout the game," said United manager David Moyes who singled out Ferdinand for particular praise.
"I'd have liked to have played better going forward. I thought we did a lot of good things in the game."
Moyes refused to accept United were out of the running for the Champions League places despite being 11 points behind fourth-placed Liverpool with 12 games to play.
"We will try and win the next game and try and pick it up. I believe the desire is still there, they are a great group of players," said the Scot.
"If there's one team that has been great at winning games in the second half of the season it's Manchester United."