Apollo 13 should count, but not for what most people know about.
The second stage center engine shut down early due to a thrust chamber sensor reading low pressure. While this did not impact the orbital insertion (the remaining 4 engines fired for an additional 4 minutes to make up for it), the sensor reading that shutdown the engine may have been in error, and is still not understood. However, if this shutdown had not occurred, the vehicle would likely have been lost in just moments after when the shutdown occurred. The center engine was experiencing severe pogo osculations, resulting in the engine flexing the thrust frame up and down by 3 inches, 16 times each second—this motion would have resulted in disintegration of the rocket in short order. The thrust chamber sensor should have been unaffected by the pogo, so that is unlikely to be the cause of the reading that led to the shutdown. So it is possible that while one failure during launch almost destroyed Apollo 13, a second failure actually (temporarily at least) saved the mission.
See this article for more information.