I went all the way back to the Senators days to check this out, and could only find six years: 1974, 1978, 1988, 1989, 1990, and 1996 where four full-time starters won ten games. The '77, '90, and '96 teams actually did have five ten game winners, but two were relievers and the other finished the season out of the bullpen, winning 9 games as a starter.
I realize I'm just begging for a course-correcting freak injury, as well as pointing at a stat most modern baseball fans consider irrelevant, but CJ Wilson (10-3), Alexi Ogando (10-3), Colby Lewis (9-7), Derek Holland (8-4), and Matt Harrison (8-7) are on the cusp of becoming first quintet of starters in team history (51st season) to post 10-win seasons in the same year.
1974: Fergie Jenkins, Jim Bibby, Jackie Brown, Steve Hargan
1977: Gaylord Perry, Doyle Alexander, Bert Blyleven, Dock Ellis, Adrian Devine (reliever, 2 starts)
1988: Charlie Hough, Jose Guzman, Paul Kilgus, Jeff Russell
1989: Nolan Ryan, Bobby Witt, Kevin Brown, Charlie Hough
1990: Nolan Ryan, Bobby Witt, Kevin Brown, Charlie Hough, Kenny Rogers (reliever, 3 starts)
1996: Ken Hill, Roger Pavlik, Bobby Witt, Darren Oliver, Kevin Gross (part-time starter, 19 starts in 29 appearances, 10th win out of the bullpen)
But since we all know wins don't matter, lets re-frame the issue. The truly remarkable thing about this season's starters, and the reason they're already so close to doing what no other set of pitchers in team history has ever done is the fact that the five of them have started 94 out of a possible 97 games, with Dave Bush getting the other three. And since Bush has been released, those five are currently the only pitchers on the roster to start a game all season.
The Rangers started 10 different pitchers in 2009 and 2010, 15 different pitchers in 2007 and 2008, and 12 different pitchers in 2006. Wins may be a devalued stat these days, but consistency is as good as gold.