In a group also featuring four-time winners Brazil and Iran, African powerhouse Senegal were forced to come from behind for victory.
They trailed 2-0 at the end of the first period, before drawing level 3-3 heading into the third and final third.
Oleg Zborovskyi scored twice for the Europeans but Fall provided the star turn netting four times, including the winner with a little over a minute remaining.
The match opened in unusual fashion with Ukraine goalkeeper Vitalii Sydorenko becoming just the sixth different goalkeeper to score at the tournament, with the Strogino Moscow No1 netting spectacularly in the second minute.
The Ukraine custodian also saved a penalty but gradually the strong and athletic Senegalese gained the ascendency on the scoreboard in a match that remained on a knife-edge throughout.