United led in the Premier League match through a goal from Antonio Valencia in first-half stoppage time at the Emirates Stadium.
Robin van Persie levelled for Arsenal on 71 minutes, but United claimed all three points when Danny Welbeck scored to make it 2-1 with 10 minutes remaining.
"We created an amazing amount of chances and to only score two is a bit disappointing," Ferguson said.
"We should have rammed home our advantage in the first half, and in the first 15 minutes of the second half they had three great chances and missed them all, so we rode our luck."
"When they put (Per) Mertesacker up and kept tossing the ball into the box, you never know what's going to happen but we coped well with that, (Chris) Smalling was fantastic in that period and we managed to see it out."
Ferguson felt the victory was particularly important after league leaders Manchester City had beaten Tottenham 3-2 earlier in the day.
"Winning after City had won was the important thing but we've done it in the right way," he said.
"We were really adventurous, positive and had a great belief in ourselves."
According to their manager, Arsene Wenger, the Arsenal players began the match apprehensively following consecutive league defeats to Fulham and Swansea.
"We had a difficult first half, we were a bit nervous due to the fact we lost the last two games and we played under pressure today," he said.
"In the second half we had chances to win the game, we made a defensive mistake on the second goal but we had a good second half and at 1-1 it looked like it would be us to win."
"We are in a fighting position, let's be inspired by the second half today."
The Arsenal fans made no secret of their displeasure when the influential Alex Oxlade-Chamberlain was withdrawn for Andrei Arshavin with 16 minutes to go.
The Gunners boss cited fitness concerns as the motivation behind the unpopular switch.
"He (Oxlade-Chamberlain) started to stretch his calf and he had given a lot, he is not used to that intensity," Wenger said.
Source: Soccerway
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