Newcastle added another score early in the second period before a fine try from scrum-half George de Cothi.
A conversion and two penalties from White put Warriors just one point behind the visitors before the fly-half's late drop-kick sailed through the posts to give Worcester the win.
Academy Manager Mark Hewitt said: "It was a cracking match. We found ourselves behind but we clawed our way back into it with a fine try from De Cothi.
"At 24-23 with just several minutes to go, we managed to get into their half and created a drop-goal opportunity and full praise to Luke White who stepped up and won us the game.
"Newcastle were always a threat and they played some good rugby but I'm delighted with how the boys responded coming from behind to finish the game off."
Hewitt insists his side have shown great improvement over the course of the season and picked out some players who have shone during the campaign.
He added: "We finished sixth with six points which is progression from last year. The players have responded well week after week this season and have shown real development in a number of areas.
"A few players have put their hands up moving forward. They have really shown that they might have what it takes to become Warriors in the future and some of the Under 18s will feature in the Cavaliers matches we still have this season.
"The challenge now is to move forward for next year. We have an exceptional bunch of Under 16s moving up so the pathway is looking really good."