United had been frustrated by a late Fulham equaliser at the weekend but they lacked the firepower to trouble Arsenal on Wednesday, with Robin van Persie thwarted by Wojciech Szczesny as they pressed.
Moyes, though, believed that a point apiece was about right following an encounter in which both sides struggled to find their rhythm.
"It was a fair result in a tight game. We had one or two opportunities to win it, but we won 1-0 at Old Trafford (against Arsenal) 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," he told BT Sport.
Given the goals conceded against Fulham - including a late Darren Bent leveller - Moyes was happy with the way his team competed.
"We looked quite compact, defended well and had a good opportunity in the first minute (when Van Persie robbed Mikel Arteta but shot straight at Szczesny). We would like to have played better overall, but defensively we were sound and, I thought, in control."
Moyes will now take his side off to Dubai for some warm-weather training before their next game on February 22, away at Crystal Palace - although he appeared 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.
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