AGoT: Attacking and winning unit dies
tl;dr


  What happens with the region?


It's empty.


  Does the defender lose control of it?


Yes.


  Does the attacker gain control of it?


No.



The long version

Combat resolution is done in four steps, completed in order:


Determine Victor
Casualties
Retreat and Routing
Combat Clean Up


Steps 1 and 2, you don't appear to have an issue with: You know who the victor is, and you know how many casualties there will be for both sides.

Step 3 (page 21) says:


  After suffering casualties, the losing army must retreat from
  the embattled area (supporting units do not retreat.)


"the losing army" is the important part here; you've already determined the victor (and thus also the loser) in Step 1, which means casualties or not you already know exactly who "the losing army" is. They must retreat.

This is emphasised in the errata:


  Q: After winning a combat, if a player would
  suffer casualties that would eliminate all of his units
  in the embattled area, does the loser of the combat still
  have to retreat?
  A: Yes.


Step 4, Combat Clean-Up, then goes on to state:


  If the combat was won by the attacker, remove any Order token the
  defender had assigned to the embattled area (if one remains) as well
  as any Power token in the area


Again, the wording is clear "If the combat was won by the attacker" (Step 1). So in addition to retreating, the defending (losing) army also needs to remove their Power token in the area.

Therefore, the answer to your first question "Does the defender lose control of it?" seems to be an unambiguous "Yes". Without an army or a Power token, the defender no longer has any means to control the area.

Which brings us to the second question: "Does the attacker gain control of it?"

From the rules, I'd have to say "No". From "Controlling Areas" (page 24):


  A House is said to control a land area when it has at least one
  Footman, Knight, or Siege Engine in the area, or has previously
  established control in that area by placing a Power token there


If all your units are dead, you obviously don't meet the first criteria. So the question is, can you have "previously established control"? Further down the same page:


  A player may only establish control when vacating
  an area with a March Order. Events that would leave an area
  vacant (such as negative effects from a Wildling victory) does not
  permit a player to establish control after the last unit is destroyed.


You never actually vacated the area: Your units died before they ever had a chance to gain control in the first place (Step 2 of combat resolution (Casualties) happens before both Step 3 where the loser retreats and Step 4 where the loser removes his Power tokens).

Again, this is emphasised in the Errata:


  Q: At what point does the winner of a combat
  exert control over the embattled area?
  A: Control over an area does not change hands
  until all stages of combat are completed. In the case
  of a completely emptied area, at no point of time
  during the combat would the area be considered
  controlled by the attacker (and so would not count
  towards his victory condition).


And like before, with neither army nor Power tokens in the area, you the attacker also have no means to exert control over the area.



Of note, in the reverse situation (i.e. the defender wins but loses all units), the attacker would still be forced to retreat in Step 3 but the defender could still retain control over the territory if he'd previously placed a Power token, since that wouldn't be lost in Step 4 if he's the victor.