Yamal struck as the tie finished 3-3, though missed a penalty in the shootout in La Roja's triumph.
Going into the game, Yamal had been hammered by Van der Vaart, who questioned the youngster's attitude and approach to his craft.
The Dutchman snapped: “I noticed some things about Lamine that started to bother me. Pants pulled too low, not putting in much effort, slightly superficial gestures...
"If you're so young, you should be happy with every minute you play for Spain. You have to prove it.”
After his spectacular goal, Yamal celebrated by pulling down his shorts.
Then after the shootout win, Yamal took to social media, declaring: “Pants down, a goal, a missed penalty and in the semi-finals hehehehe Let's go Spain!”
Yamal also posted a series of photos to Instagram, which included one of Van der Vaart.
In response, Barca and Spain teammate Ferran Torres posted: "Lower them a little more please," along with a heart and an emoji of raised hands.
Héctor Fort was more direct: "Keep talking, keep talking."