If an attack does not do damage beyond a character's soak, but there is enough advantage to trigger knockdown, or disorient, or other qualities, can they be applied to the target?

For example a PC is brawling with a Target. The target's soak is 7. On a successful attack the dice net 2 advantage and one success so the attack hits, however there is not enough damage to get through the target's soak. Could the PC knock the Target prone or disorient them in this exchange?

From my understanding, yes. The attack only needs to hit, doing damage is not required.

As long as an attack successfully hits, you can activate a quality, unless it specifically says otherwise. Damage does not need to exceed soak to activate them.

Unless I'm mistaken, yes, the abilities can still be triggered as long as the attack hits.

Systematically speaking, you can argue that other qualities, like Blast, can still be triggered on a miss, so a hit with no damage is still valid.

Narratively speaking, you could state that the knockdown was caused by catching the target's armor and knocking them off balance, but the hit and the fall didn't do anything more than inconvenience them.

If you've seen the movie Serenity, we see a good example of this: Inara lit some "incense" which was, in reality, a flash grenade. It could have potentially dealt some damage due to concussive force, but it didn't (whether due to it being a weak blast or the Agent have a really high soak). It did, however cause Knockdown and Disorient on the Agent.

Just a thought for you to better grasp the logic there.