The Canaries have confirmed that the Scottish international midfielder has signed a two year deal at Carrow Road, with reports suggesting the fee is in the region of £3m.
Speaking about the news, the former Livingston star told Canaries.co.uk, "I'm delighted and pleased to get it sorted and glad to join permanently.
"It's been a great few months, the boys welcomed me and it's been great to work with the gaffer and I look forward to the next couple of years."
Dorrans actually began the season as a regular in Alan Irvine's Baggies team, starting 19 Premier League games. However, he was deemed surplus to requirements shortly after Tony Pulis arrived at the club and joined Norwich on loan in February.
The 28-year-old went on play 15 times in the Championship as Alex Neil's side secured third place, before featuring in each of the playoff semi-final legs against fierce rivals Ipswich, as well as coming on a substitute against Middlesbrough in the final itself.
Neil, who was only 33 years of age when he was appointed by Delia Smith and the Norwich board in January, is already looking forward to the challenge that lies ahead for the club in the Premier League.
Despite retaining a large number of the players that were relegated in 2014, the manager is looking to add an extra level of experience and hopes his fellow countryman can provide that, with more than five top flight seasons under his belt.
"Graham has been a very important part of the team over the last few months and he has Premier League experience and is a top player," Neil commented.
"His experience and quality will help us and we're delighted he has joined us on a permanent basis."