Van Persie scored the only goal of the game with a terrific header from a Rooney corner, while the Englishman was central to United's play over the course of the afternoon.
Per Mirror Football's David McDonnell, Moyes told reporters:
Great players score great goals in big games and Robin is a great player. I have to say his sidekick, Wayne, played great as well. The two of them were fantastic. Wayne made it, Robin scored it.
I said to Wayne afterwards that his energy and the way he worked was magnificent. It was a big game to win because Arsenal have been in such great form. We showed how important it was by how well we played in the first half.
Manchester United were champions last year and a lot of people wrote them off very quickly, which is fine, but hopefully by the end of the season we'll not be too far off.
United were not particularly attacking in their approach against league leaders Arsenal, but they enjoyed much the better of the game for the first hour as the visitors looked to be feeling the effects of their tough schedule.
Japan's Shinji Kagawa linked well with the front pair, while Antonio Valencia also helped the side press Arsenal back from the right. Playing in midfield, Phil Jones also had a strong early impact.
However, it was to the individual quality of Rooney and van Persie that United turned for goals. Van Persie's header across goal from a corner for the opener was sublime, while Rooney almost added a second as he screwed a left-footed effort narrowly wide after good work at the edge of the box.
Moyes was also right to commend the England man on his work rate, with Rooney's efforts in closing down Arsenal defenders in the final minutes important to helping United withstand the pressure that their opponents had begun to impart.
The pair have been massively important to United's attempt to stay in touch with the league leaders as they settle into life under Moyes. Van Persie now has seven league goals to his name, while Rooney has five goals and four assists in the same competition.
Having worked so hard to retain Rooney in the summer, Moyes will be delighted at the England man's response and the pair's increasing understanding of each others' games—something which was not always the case last year.
United are unbeaten in nine following a shock defeat to West Brom and have some tough clashes coming up in both domestic and European competition.
If Rooney and van Persie keep coming up with the goods when required, United will hope to place themselves in a good position heading into the Christmas and New Year calendar from which point in the season they have often accelerated away from their rivals.