The Dutch attacker was the Premier League's top scorer and was named PFA Player of the Year in his final season with Arsenal, but the top-flight crown continued to elude Arsene Wenger's team.
Van Persie was lured to Old Trafford in 2012 and helped fire United to the league title in his first year, again finishing as the division's top scorer. His former side, meanwhile, had to settle for fourth place.
And Frimpong believes Van Persie's departure came as a huge disappointment and exposed the club's weakness when it comes to keeping their strongest performers.
"I think the problem for Arsenal has always been how do they keep hold of their best players? It’s a problem with Arsenal," Frimpong told MetaRatings .
"I hope that trend has finished with Arsenal, because when Robin left, he was probably the best player in the Premier League at that time, and it took a lot of quality out of the team and Arsenal suffered again.
"That was the problem with Arsenal at that time; that we kept losing our best players and it was disappointing.
"When Robin was at Arsenal that year, he was fantastic – he was the captain, he was the top goalscorer, he was the best player in the Premier League. And for him to go to Man Utd was quite disappointing.
"The club knew better, and I don’t know what goes on behind closed doors, but I think there was a bit of a problem.
"I listened to an interview with Robin a couple of weeks ago, and he spoke about the situation at the time, you just have to respect the man in his decision.
"Sometimes these things work like that. You just have to wish him well because he served Arsenal very well when he was there."
A year before Van Persie's move, Arsenal saw another star player leave as Cesc Fabregas joined Barcelona.
Although the Spanish midfielder had been the captain for three seasons, Frimpong believes Van Persie's exit had a bigger impact on the north London outfit.
"I think I was more disappointed when Robin went to Man Utd because at that time he was the captain and he was the best player in the Premier League," he added.
"And losing Robin was much more of a blow than losing Cesc because Robin was bringing goals to the game, he was winning us games.
"He was playing really well for us at the time so I feel like he was a big blow at the time, and more than when Cesc left. These things happen in football, and you can’t control it. It’s something that is difficult for the club when you have players who are on less than a year’s contract, they want to leave and you want to make money; it makes sense."
However, the 27-year-old doubts that Wenger's team would have been able to win the Premier League title had they held on to Van Persie.
"To be honest, and as an Arsenal fan, I don’t think so, because the players we had at the time were not good enough to win the Premier League," he said.
"Man Utd had the squad; they had the players. You can’t win the league on Robin van Persie alone. They had Rio Ferdinand, [Wayne] Rooney was there at the time, and they had quality players to win the Premier League. Even as an Arsenal fan, we didn’t have that quality.
"We had good players, but they had great players, and even if Robin had stayed I don’t think we could have won the Premier League."