The 21-year-old - who swapped Zwolle for his new club in July - took advantage when goalkeeper Robbin Ruiter went up for a stoppage-time corner, the attacker lifting into an empty net from more than 60 yards for his first goal in professional football.
That came after Jesper Drost had given the hosts the lead as Zwolle followed up their Johan Cruyff Shield victory over Ajax with three points in the first game of the new Eredivisie season.
For Utrecht, though, Friday's result represented a poor start to Alflen's tenure, with the head coach having succeeded Jan Wouters following a disappointing campaign last time out.
Defences were on top in the first half as Zwolle's repeated attempts to exploit the wide areas proved fruitless, with Jody Lukoki alone providing 10 crosses that failed to produce a breakthrough.
But Zwolle, who begin their UEFA Europa League campaign against Sparta Prague later this month, were rewarded for their persistence 15 minutes from time as Drost fired home.
The ball fell for the midfielder at the back post from Lukoki's right-wing delivery and he found the bottom-right corner from the edge of the six-yard box.
It got worse for Utrecht, who have now failed to keep a clean sheet in 25 consecutive away league games, when Moro audaciously stole the show with his last-gasp moment of brilliance.
