Grande, who pulled out of a planned performance at the ceremony, received best pop album for Sweetener - her first ever Grammy Award.
"This is wild and beautiful," she tweeted. "Thank you so much."
Gaga won best song written for visual media, for the A Star Is Born track Shallow; and best pop single for the ballad Joanne.
Stars are still arriving for the main show, which starts at 0100 GMT.
Among the first to walk the red carpet was Camila Cabello, who will open the ceremony later.
"The performance is based off of my grandma's childhood, and I have my family in the performance," she said on the red carpet.
"I definitely have severe butterflies."
The show will also include a tribute to Dolly Parton, with the country star joined on stage by Katy Perry, Kacey Musgraves, Little Big Town and Maren Morris.
"I'm just internally freaking out," said Morris before the show. "I met her and I was just trying not to quote Steel Magnolias at her."
Other early awards included The Greatest Showman, which won best soundtrack; and best music video, which went to Childish Gambino's incendiary This Is America.
Dua Lipa won her first Grammy for the Silk City collaboration Electricity, which got best dance recording. The British star is up for best new artist later in the show.
"I'm so emotional," said the British star, who is also up for best new artist in the main ceremony. "I keep trying to look after my make-up. It's watering on the sides!"
Fellow Brit Ella Mai also won an early trophy - best R&B song for the slinky summer hit Boo'd Up.
"I'm legit trying to take it all in," said the singer, who's also nominated for song of the year. "I've dreamt of this moment ever since I was a little girl."
Is Quincy immortal? At the age of 85, country star Willie Nelson won best traditional pop vocal album for My Way. He's far from being the oldest-ever Grammy winner, though. That accolade is held by Pinetop Perkins, who won best traditional blues album in 2011, when he was 97.
Brandi Carlile and Kacey Musgraves swept the boards in their respective genre categories, Americana and Country, making the ones to beat in the night's main category - album of the year.
And Quincy Jones was honoured in the best music film category, for the Netflix film Quincy.
The award was accepted by his daughter, Rashida Jones, who said: "When he first saw the film he said, 'I wish I could live forever - and I think he could be the first person to do that. I hope you do, dad."
Kendrick Lamar and Drake lead this year's nominations, with eight and seven respectively.
They are both up for album of the year, alongside Cardi B, H.E.R., Post Malone and Janelle Monae.