The left-hander pitched his first career shutout in Tuesday's 8-0 win over the Rays. He scattered five hits, issued three walks and recorded six strikeouts.
He followed that up by pitching eight scoreless innings in the Rangers' 8-1 win over the A's on Sunday to give him a career-high 16 victories.
The 30-year-old is the first Rangers southpaw to garner the weekly honor since Kenny Rogers tossed a perfect game in 1994, and Wilson joins Michael Young as the second Rangers player to take home the award this season.
In a league-leading 31 starts, Wilson has a 3.01 ERA with a 1.172 WHIP (walks and hits per inning pitched).
The 17 scoreless frames is the longest streak since Wilson became a full-time starter prior to the 2010 season.
In a Sept. 1 start vs. Tampa Bay before the streak began, Wilson took a perfect game into the fifth inning, but he lost his command when a ground ball deflected off his index finger.
In seven starts since Aug. 12, the fifth-round pick in the 2001 First-Year Player Draft has a Major League-best six wins (6-1 overall) with a 1.89 ERA, 44 strikeouts and 12 walks in 47 2/3 innings.
The Rangers plan to start Wilson every fifth day for the remainder of the season, so his next start is Friday against Seattle.
Louie Horvath is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.