I can see your point as Gandalf and The Balrog are responsible for each-other’s deaths.
So, in Gladiator, does Maximus not win his final fight against Commodus because he dies after from wounds sustained from Commodus?
If the aim of a duel is to defeat your opponent, the loser would surely be defined as the one to die first.
Therefore, Gandalf can still be considered to have won the battle despite the fact he basically rolls over and dies immediately after throwing down his enemy.
Succumbing to your wounds after the fact, irrespective of the timeframe, wouldn’t change the outcome of the initial contest.
Definitely an interesting topic though- which unfortunately has no place here.