I just checked, and the situation with the 1987 Redskins' QB position was pretty crazy.
Jay Schroeder started 10 games (all in regular season) and the Redskins won 8 of them. Ed Rubbert started 3 games (all in regular season) and the Redskins won all 3. So, two different QB started games in the regular season which the team won.
Doug Williams started two games in the regular season, and the Redskins lost both. However, he came in as a replacement in 3 games, making more attempts than the starter in all 3, and the team won all 3 of those.
Moreover, in one of the games Rubbert started, he made only 2 pass attempts before being replaced by Tony Robinson, who finished the game (with 18 attempts), leading the Redskins to a win.
So, you can make a good case that this Redskins team really won games during the regular season with 4 different QBs.
In the playoffs, Williams started each game, and the Redskins rolled to a victory each time, beating the Broncos 42-10 in the Super Bowl. Schroeder was credited with one pass attempt (which resulted in a sack) in that SB victory.