Veteran Sweden defender Mellberg, who headed to Athens from Juventus in 2009, knows all about the Dutchman from his seven seasons at Aston Villa.
Van Persie chalked up his century for the Gunners in the Barclays Premier League win over Bolton at the weekend.
Olympiacos will have to be on their guard to avoid what would be a 10th successive defeat on English soil - their run including a 2-0 loss at Emirates Stadium in September 2009, when van Persie was on the scoresheet.
Mellberg accepts the 28-year-old World Cup runner-up has developed into one of the most dangerous forwards around.
"I have played against him a few times, but maybe not in that position. He plays mostly as a striker now from what I have seen. He was more of a wing player when I played Arsenal with Aston Villa," Mellberg told a news conference at Emirates Stadium.
"He is a good player and they have a few more.
"We will focus on our own game more obviously, but we will look at them tomorrow [Wednesday] and hopefully will be prepared for him and everybody else they have.
"I have obviously heard about some problems they have had and they haven't had the best of starts in the league, but it is always a really difficult game, especially here at Emirates Stadium."
Olympiacos lost their opening Group F game at home to Marseille, which was their first competitive match of the season after two domestic Super League matches were postponed when the teams they were scheduled to face - Kavala and Olympiacos Volou - were relegated because of a match-fixing scandal.
Coach Ernesto Valverde is hoping for an improvement on Wednesday.
He said: "We feel better compared to the previous European match day given that we have had some official games beforehand.
"However, this does not mean that tomorrow's [Wednesday] game is an easy one for us.
"We have to face an opponent who is probably one of the toughest we could find ourselves against.
"We know that it will be very difficult, but we came here to take something out of this game and will do our best."
For Arsenal midfielder Mikel Arteta, this season is a first taste of top-flight European action following his deadline-day switch from Everton.
The Spanish midfielder, 29, is relishing the challenge.
He said: "It is hard to explain what it is about the Champions League, the atmosphere is different you are playing against the best teams in Europe every time.
"You are expected to win here, which is a really good thing.
"We need to be prepared for that and have to take things game-by-game. We all know we have the ability to do it."
Arsenal will be without half-a-dozen first-team regulars for Wednesday night's Group F clash, including England midfielder Jack Wilshere.
Forward Theo Walcott, Ivory Coast striker Gervinho and centre-back Laurent Koscielny have also picked up minor niggles after Saturday's 3-0 Barclays Premier League win over Bolton.