The world number 35 edged the first set but hit his peak in the second and left the Dutchman no chance in Malaga, winning in 76 minutes.
World number one Jannik Sinner faces Tallon Griekspoor in the second singles rubber aiming to seal a second consecutive triumph for his country.
Berrettini, who came to support Italy as they ended a 47-year wait for the trophy in 2023 as he missed out with an ankle injury, has been crucial in their Davis Cup defence.
The 28-year-old paired up with Sinner to help Italy triumph in a doubles decider against Argentina in the last eight, before winning his semi-final singles rubber against Thanasi Kokkinakis.
"It feels unbelievable, it feels really great to be on court -- if you win it's better, but being able to be healthy, here and enjoying this kind of atmosphere is the most important thing for me, because I missed that," said Berrettini.
"I missed these moments and that's why I'm really happy. We still have hopefully just one more match to go, but the first step was taken."
Van de Zandschulp, ranked 80th, became the last player to defeat retired Spanish superstar Rafael Nadal earlier in the week as Netherlands reached the final for the first time in 104 years of trying.
However he was blown away by Berrettini, who moved ahead in the ninth game of the first set.
The Italian opened up three break points and took the first to claim a 5-4 lead when his opponent pushed a shot into the net.
In the second set Berrettini, who finished as Wimbledon runner-up in 2021, broke in the third game and again in the seventh.
The Italian produced 10 forehand winners and 16 aces on his way to victory, clinched on his second match point.
