Van Gaal and Cruyff played together at Ajax and later both men managed the Dutch club, while Van Gaal also took charge of Barcelona a year after Cruyff left the Catalan side.
"It is a very sad day for the footballing world, to lose one of the true legends of the game," Van Gaal told ManUtd.com.
"My thoughts are with Johan's family at this difficult time."
Manchester United winger Memphis Depay also paid tribute to Cruyff ahead of the Netherlands' match against France on Wednesday.
"Recently I heard someone say, 'When I'm abroad and tell people I'm from Holland, they say the country that's famous for its tulips and for Johan Cruyff.' That's something that will always remain the same," said the winger.
"A true legend, who's no longer with us. It's a sad day."
Former Manchester United players and Netherlands internationals Peter Schmeichel and Robin van Persie posted on social media following Cruyff's passing.
Schmeichel posted on Twitter: "Terrible news that Johan Cryuff has passed away. One of the greatest players ever, a true giant in our game, he will be sadly missed. RIP."
Terrible news that Johan Cryuff has passed away. One of the greatest players ever, a true giant in our game, he will be sadly missed. RIP— Peter Schmeichel (@Pschmeichel1) March 24, 2016
Van Persie added on Instagram: "RIP Johan Cruyff. The best ever dutch player we had and will have! The man who changed the game we all love! You have been a big inspiration in my football life Johan! Thank you and may your soul rest in peace!