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Van Gaal took the bold decision to remove captain Van Persie when Holland trailed Mexico 1-0 and Klaas-Han Huntelaar, his replacement, set up Wesley Sneijder for the equaliser and scored the injury-time winner from the penalty spot.
Van Persie, who will link up with Van Gaal again at Old Trafford when the 62-year-old's reign as Manchester United manager begins in earnest, believes his compatriot got his tactics spot on.
He told reporters in Brazil: "This trainer wants to win. So he makes substitutions. It's simple.
"We played three different formations, so it's a big compliment to the players that they picked up on it so quickly."
Netherlands have won all four of their games in the World Cup, beating holders Spain, then Australia, Chile and Mexico to set up a last-eight tie against Costa Rica.
And Van Persie feels they have surpassed expectations already but thinks they can go on to become the first Dutch team to win the tournament.
He added: "Let me tell you, the Dutch media thought we would never survive the first round, so how we ended up becoming favourites for the last eight, I'll never know.
"This group of players is very modest. We have an outstanding spirit, an outstanding atmosphere, that led to this victory. Everyone feels we can win it, that it really is possible.
"To be the captain of this squad makes me incredibly proud. It's a huge honour to be here, at this stage, wearing the armband, because we are a real team.
"All 23 players are important and also the technical staff. The way they prepare for competitions is unprecedented. I've never before known a level like it."