At pitched battle at the end the rebels continued knocking Vader down but to have failsafe keep Vader alive. We are wondering for failsafe to activate Vader must for a split second actually be defeated right? If that is the cause since our mission was to defeat Vader, would the players win? For the order of resolution is mission (kill vader), Imperial Player (activating failsafe), and rebel players (nearly hanging the Imperial player), or does failsafe activate first?

From reading the text of the card, it says "Instead of being defeated, that figure recovers 1 damage and gains 3 movement points." Going by that text, Vader is never technically defeated, he remains undefeated with 1hp remaining.

The annoying thing is that he doesn't go down, the benefit is that he has to exhaust the card and use two threat. If he has no threat he can't use it, and if it is already exhausted, he can't use it. All you have to do then, is just put a point of damage into him.

Note the new fixed wording of Failsafe in the FAQ.

When a figure has suffered damage equal to his Health, the figure is defeated.

When a figure has suffered damage equal to his Health, you can trigger failsafe as long as you can pay the costs (threat and exhaust). (Failsafe does not trigger from a figure being defeated, but from having suffered damage equal to his health.)

The resolution order (see Timing, page 2 of RRG) for abilities with simultaneous triggers would otherwise be mission rules, imperial, rebel, but Failsafe explicitly tells to perform it first.

If you trigger Failsafe, it is resolved first. The you check the defeat condition. Because Vader no longer has suffered damage equal to his Health, he is not defeated.

That is how we played it out, just wanted to make sure. Thanks so score one victory for the imps.