It might be difficult to figure out, but that list seems to be pitcher wins in a row without recording a personal loss, so leaving as not the pitcher of record and the team losing wouldn't count against you.
"Clemens needed a little luck along the way, escaping with a no-decision in seven different starts"
The streak priester is on feels a little more special because even if he doesn't get credit for the win, the team still does win.
Using the article as a springboard, looks like it might be Arrietta with 23 followed by Hubbell with 22 if you don't include a game he pitched the final 5 frames or 23 if you do include it. Since Quinn has had a couple where he came in the 4th inning, I'd say count it, but its still getting pretty close unless there was another who just wasn't lucky at picking up the wins. Btw, I believe 3rd on the list was 15 so he's passed that.