Robin van Persie had the best two chances for United, but fired tamely at Wojciech Szczesny in the first 90 seconds and then saw his strong header well saved by the Pole late on.
Arsenal pushed for a winner in the closing exchanges but neither side threatened for prolonged periods, and Moyes had no complaints with the outcome.
"It was a fair result in a tight game," he told BT Sport. "We looked quite compact, defended well and had a good opportunity in the first minute. We would like to have played better overall, but defensively we were sound and, I thought, in control.
"We had one or two opportunities to win it, but we won 1-0 at Old Trafford earlier in the season, and if you said at the beginning of the season we would take four points off Arsenal, I would have taken that."
Moyes will now take his side to Dubai for some warm-weather training before their next game on February 22, away at Crystal Palace - although he was reluctant to discuss the details of the trip.
"We don't usually talk about that. It will be a mixture of hard work and relaxation," he added.