The succinct answer, at all levels of play—high school, college, pro, softball, baseball—is yes.
Official Baseball Rule 5.06(b)(3)(C) Comment, for instance, states, "If a fielder, after having made a legal catch, should step or fall into any out-of-play area, the ball is dead and each runner shall advance one base, without liability to be put out, from their last legally touched base at the time the fielder entered such out-of-play area."
The minutiae of the rules are different, but largely agree: If a fielder establishes firm and secure possession in their hand or glove of a ball in flight prior to falling out of play, it is a catch ("the act of a fielder getting secure possession in their hand or glove of a ball in flight and firmly holding it"). What this effectively means is that, because the fielder in this play appears to catch the ball while still airborne with feet over the fence and not in contact with dead ball territory, this is a legal catch (OBR 5.09(a)(1) adds the requirement, "must have one or both feet on or over the playing surface").
If the fielder were to have juggled the ball while falling to and contacting the ground, or had the ball fallen out of the glove when the fielder's elbow/back hit the ground in dead ball territory, this would not be a catch: that's what makes this call tricker than it appears on the surface. The umpire must determine whether the fielder legally caught the ball (gained secure possession) prior to falling out of play.
The outcome for a legal catch and carry is a dead ball, batter declared out, and any runners awarded one base from time-of-pitch. If the umpire determined this was not a catch, the outcome would be to award the batter a home run.
Video as follows:
Alternate Link: Utah HS softball player's catch at the fence: what's the rule? (KSLSports.com/CCS)
