Ajax had their chances but failed to convert them as Fabinho produced a 'Man of the Match' performance after slotting in at the back following Virgil van Dijk's knee injury over the weekend.
But Liverpool's away win failed to convince Van Persie who said they clearly missed Van Dijk and Thiago.
Speaking after the match on BT Sport, he said: "They clearly missed their main players. Virgil at the back of course is a big miss, Thiago - although he didn't play a lot - is a world-class player.
"But Ajax had the bigger chances tonight. They played quite well, they had four or five clear chances, and normally they actually take one of them.
"Not tonight. So I think Liverpool have been lucky tonight."
When asked if Ajax deserved something from the game, he replied: "Yeah, I believe so.
"Ajax did well, created many chances, but didn't get their goal."
Van Persie, though, did hail the performance of Mane and praised Jordan Henderson's contribution.
"For me, in the first hour, it was a joy to watch Mane. His movement, the way he holds up the ball, his link-up play - I would pay to watch him! He's a fantastic player to watch," he added.
"You could see Liverpool were searching in the first half, but when Henderson came on you could see his leadership.
"He was yelling to his teammates to go out on every single duel, he was really good on the ball, confident. He was like a proper, mature player, a real captain, so he made a big difference as well in the second half."
Liverpool resume their Premier League campaign on Saturday when they host Sheffield United.
Featured Image Credit: BT Sport/PA
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