Van Persie, who joined from Arsenal for £24 million late last week, was introduced with 22 minutes remaining as United chased an equaliser.
The Netherlands striker did not register a shot on target as the Red Devils lost 1-0, although he almost set up fellow new signing Shinji Kagawa with a squared pass.
"We played around him too much, we didn't show enough penetration," said Ferguson.
"With Robin in your team you know you want to use his ability against centre-backs in situations he is very good at - but we didn't use him enough."
His headed goal - while being marked by Carrick - on 57 minutes sent the Toffees fans inside Goodison Park into raptures.
Ferguson was scathing of Everton's tactics in targeting the attacking midfielder.
"He (Fellaini) is a handful: he is a big, tall, gangly lad and they just lumped the ball forward to him, that's all they did," added Ferguson.
"They worked from that base all the time and they got a goal from him, so it's justified."
Everton manager David Moyes said that there is still plenty of work to do if they are to rid themselves of their 'poor starters' tag.
It was the club's first triumph in their opening fixture for five seasons.
"We haven't started seasons well and even (this) result doesn't tell you that you have started the season well," he said.
"Just as if we had lost we wouldn't say we had started the season badly. You probably judge it after 10 games - how you've started the season.
"I've tried absolutely everything to try to get off to a good start so we tried something different this year.
"I am really pleased because they performed ever so well and got a great result."
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