Mr Norris received the most first and second preference votes, with a total of 125,482.
He beat Conservative candidate Samuel Williams, who received 85,389 votes, by 40,093 votes.
The seat, created in 2017, was previously held by Conservative Tim Bowles, who did not seek a second term.
Mr Norris praised Labour leader Sir Keir Starmer after the results were declared at the SS Great Britain in Bristol on Saturday.
“I’m very proud of our region, it’s a great region but it could be even better,” Mr Norris said.
It s a huge honour to be elected. Thank you. I will show the difference an active Metro Mayor makes for our region. This role is about collaboration and I ll listen carefully and work cooperatively. pic.twitter.com/oBvYBgRqjH