US international Josh Sargent gave the Canaries a first-half lead when his shot flew through the hands of keeper Alex Palmer after he ran onto Kenny McLean’s perfectly weighted pass.
Having failed to seriously test Angus Gunn in the first 45 minutes, Carlos Corberan’s Baggies raised their game after the break but a flowing Norwich move ended when Dimi Giannoulis squared to Rowe and he swept the ball home.
Unmarked Daryl Dike headed straight at Gunn as the visitors looked for a consolation, but Norwich held out and their eighth clean sheet of the season extended their unbeaten run at home in the league to six matches.
The win also lifted them three places to eighth in the table, just two points outside the top six and only four behind West Brom, who remain fifth.
Norwich boss David Wagner made six changes to the team which began their midweek FA Cup replay win over Bristol Rovers, while Albion recalled Jed Wallace in midfield, with loan signing Andi Weimann starting on the bench.
The home side began brightly and went in front when McLean carried the ball forward after winning it in his own half, threaded a pass between two defenders and Sargent did the rest.
Weimann was sent on for the second half in place of Adam Reach and he soon picked out a good pass to John Swift, whose shot was deflected away for a corner.
Thomas-Asante failed to get enough power on a header to beat Gunn from Swift’s cross and was replaced by Dike soon after – but the striker had only been on the pitch for a couple of minutes when Rowe again showed why he is reportedly attracting interest from Premier League clubs with the second goal.
After failing to beat Gunn from Alex Mowatt’s cross, Dike then cleared the bar after Weimann nodded down to him before Rowe brought another good save out of Palmer after twisting away from his marker as Norwich saw out the game to secure their first win in six league matches against Albion stretching back to 2007-08.