Fellow New Zealander Chris van der Drift also bagged a class podium, finishing fifth overall.
Teenager Armstrong extended his lead in the Italian Formula 4 Championship after winning the first race of the weekend at Imola.
The Toyota Racing Series star had a fierce battle with Estonian teammate Juri Vips but came out on top as Vips veered off the circuit.
He went on to claim a solid win and consolidate his championship lead.
After suffering from a steering rack problem in qualifying Armstrong started races two and three from eighth on the grid.
He produced a stunning drive to make his way to fourth in race two while he managed one better with third in the final race.
He takes a 17-point championship lead to the final round at Monza next month.
But Nick Cassidy suffered a disappointing weekend in the Japanese Super Formula with a DNF at Autopolis in Ōita Prefecture.
Cassidy crashed midway through the race.