You would not provoke any Opportunity Attacks using this method (and neither would the hand)
The PHB tells us when an opportunity attack is provoked:
You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.
So you are correct in saying that the unconscious carried creature would not provoke any Opportunity Attacks from enemies whose reach it leaves because it is not using movement, actions, or reactions to move. So enemies can't lay a finger on you in this way.
Thus, this plan works as you say.
Side note: RAW Bigby's Hand cannot provoke OAs
I realize this is a bit nonintuitive, but as written there is no way for the hand to provoke OAs because it is not a creature. Per the description of the Bigby's hand spell (or arcane hand, as it is listed in the SRD):
The hand is an object...
And only creatures provoke OAs, per the description of opportunity attacks:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
Even if you houseruled that the hand could be handled as a creature, it still would not provoke OAs because of the rule we cited earlier. Since the hand is not moving using its action, reaction, or movement to relocate because the caster is using their bonus action to control and move it.
When you cast the spell and as a bonus action on your subsequent turns, you can move the hand up to 60 feet and then cause one of the following effects with it.
It is your bonus action that you spend to move the hand. The hand does not even have any actions or movement of its own that it can use.
So, unintuitive as it may seem, the hand cannot provoke OAs because it is an object and because it doesn't use its own actions or movement to move.1
So, yes the technique works and avoids OAs against the carried creature and the hand and is a clever and handy strategy to use in cases like the one in question.
1 - Note: Even if you houseruled that the hand provokes OAs (despite all the rules reasons that it should not), the attacks would have to be directed at the hand itself and not at the carried creature. There is no way by default to redirect an OA at another target that did not provoke an attack.