In total, the Barons had five first-round selections in their lineup Friday plus another seven players selected in the second round (none lower than 48th overall).
But they couldn't start the season off with a win — not even with a last-minute power play and a sixth attacker.
After a scoreless penalty-marred opening period that saw the Monsters outshoot the Barons 12-10, Lake Erie's Bill Thomas, who played seven games with the Florida Panthers last season, got the game's first goal, putting away a puck that went in off Yann Danis' right pad.
Schultz, who was impressive in his pro debut, picked up an assist when he sent a quick feed over to Mark Arcobello on a 5-on-3, leaving the game in balance until 16:27 when Agozzino flicked the winner over Danis' right shoulder.
The Monsters and the Barons face off again on at noon Saturday, before the Oilers' AHL affiliate returns to Oklahoma City for its Oct. 19 home-opener against San Antonio in the Cox Center.