The One Direction star will play 12 nights at the venue in a bid to beat Coldplay's record of 10 performances there last year.
The string of shows is poised to smash the record for the most performances by any artist in a single year at the national stadium.
Styles will also beat pop superstar Taylor Swift's record for the most concerts performed at Wembley by a solo artist during a single concert run.
She performed eight shows at the venue as part of the Eras Tour in 2024.
In a tribute to Hockney, whose death at 88 was announced just hours before his show, Styles flashed up a quote from the artist on screens around the venue before performing his hit Aperture.
It read: "What an artist is trying to do for people is bring them closer to something, because of course art is about sharing.
"You wouldn't be an artist unless you wanted to share an experience, a thought."
Hockney painted the pop star in a work displayed at the National Portrait Gallery in 2023.
The portrait depicted the singer wearing an orange and red striped cardigan, blue jeans and a pearl necklace.
Since news of his death was announced, tributes have poured in for Hockney, one of the most significant and influential artists of the twentieth and twenty-first centuries.
Styles skipped out on stage to delirious screaming as he opened the show with Are You Listening Yet? from the album Kiss All The Time. Disco, Occasionally.
He said: "Good evening, my name is Harry.
"It's an absolute pleasure to play for you tonight.
"Our job is going to be to entertain you, I promise you we will do our very best.
"Feel free to be whoever you want to be."
He told the crowd to hold hands and take a collective deep breath, before adding: "Now let's have some fun."
During the show Styles recalled that the venue is right next door to the location of his X Factor audition 16 years ago.
He said: "My sister brought me here to London for the very first time for my X Factor audition.
"It was right in that building next door that I was put in a band.
"Driving here today I went the same way I went when I first went to that building.
"My sister is here tonight and I want to say thank you."
To his fans, he added: "It means a lot for me to be here tonight. Thank you so much for allowing me to do these shows. It means so much."
He added: "It's an absolutely pleasure to be here at Wembley stadium. I'm aware I don't get to do this without you guys."
Over the course of an energetic two hours, he performed hits including Watermelon Sugar, As It Was, Sign Of The Times and American Girls.
Styles kicked off the Together, Together Tour in Amsterdam.
On June 16 he will perform at the Royal Festival Hall as part of the Meltdown Festival, which he curated.
Country star Shania Twain opened the show, performing hits including Still The One, That Don't Impress Me Much and Man, I Feel Like A Woman.
13°C