It remains one of the league's most famous strikes as the then 16-year-old Evertonian turned and drove the ball beyond David Seaman to end Arsenal's 30-match unbeaten run and announce himself at the highest level.
Rooney scored another 16 times for the Toffees in 77 games before moving to Manchester United in August 2004 for £20million.
Reflecting on the striker's time at Goodison Park, Moyes said: "We knew what we had at the club. I got told about him as soon as I came in the door.
"He was everything we hoped. He was a man when he was very young. He had the ability and the talent but also a desire to play football.
"Wayne could have been a goalie, he could have been a right-back, a centre-half, he could play any position because it was his love for football which I think was one of the biggest factors.
"He didn't need coaching because when he came in he could play with good players, he adapted to what he had to do.
"I think you come across these players very rarely. England have had some great players over the years but I don't think you get batches and batches of them, and Wayne is one of them."
Rooney has had plenty of disciplinary problems - his first red card followed soon after his first goal - and Moyes described his former charge as a street footballer.
The Everton manager said: "All he wanted to do was get the ball. There was none of this warming up and making sure you're all stretched, not with Wayne, it was a case of, 'I'm on the street and I'm going to go and kick the ball and do what I do'.
"It was a throwback to the way I felt I was brought up. Wayne was a street player and he showed that when he came out onto the training ground as a professional. He was a joy to have around - he was hard work at times but he was a great player."
Moyes added to his ranks on Friday with the signing of free agent Thomas Hitzlsperger on a short-term deal until January, subject to international clearance.
The former Aston Villa and West Ham midfielder has been training with the club in a bid to earn a deal after leaving Wolfsburg in the summer.
Hitzlsperger, 30, managed only two league starts in the Bundesliga last season after suffering a knee injury and Moyes admitted he may not even play for the Toffees in the Premier League.
The Scot said: "I'd be careful of using the word signing when you bring in someone for a couple of months. We wanted to have an extended look at him. He's already been here for about a month.
"He needed the training because he's missed a lot of football. He's been a great lad to have around the place and we're more than happy to give him that opportunity.
"We're not saying what he's going to bring to us. He may come here and not play, we're not sure yet, we just want a longer period to have a look at him, make sure that he's fit and he can offer us something."
Moyes has a few problems in midfield for Sunday's trip to QPR, where he will be without Marouane Fellaini and Darron Gibson.
Fellaini, who has been one of the most impressive players in the Premier League this season, picked up a knee injury against Wigan two weeks ago but Moyes hopes he could be back for next weekend's Merseyside derby.
That could also be the case for defender Tony Hibbert (calf) but Gibson is taking longer to recover from a thigh problem.
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