Connette, a junior from Corona, Calif., had 324 yards passing and 55 rushing. The five touchdowns are a career high for Connette, who threw for four scores in last week's 58-55 loss to Pitt.
Josh Snead also had 108 yards rushing for the Blue Devils (3-2), who snapped a two-game losing streak with the victory.
A holding penalty negated a 56-yard touchdown catch by Troy's Chandler Worthy with 9:37 remaining that would have given Trojans the lead.
While the unit had its troubles against the Trojans, it only allowed three points in the final 28-plus minutes, forcing a number of three-and-outs and not allowing a big play in that span.
Before then, the Blue Devils built two separate 14-point leads before allowing Troy back in the game.
After Chunn scored on a 1-yard run shortly before halftime, Robinson connected Wilson Van Hooser for a 40-yard touchdown reception to tie the score at 28 with 13:15 left in the third quarter.
Duke converted on three different third-and-longs on its next drive, retaking a seven-point lead when Connette spun his way into the end zone from 4 yards out with 9:46 remaining.
The Trojans pulled within four points — 35-31 — when Will Scott hit a 37-yard field goal with 12:44 remaining in the game.
But the Blue Devils' defense stiffened from there, and Troy's best offensive play — the 56-yard reception by Worthy — was called back.
Duke's Ross Martin kicked a 34-field goal with 31 seconds remaining to give the game its final score. Troy got the ball on its 22-yard line with a chance to drive for a game-tying touchdown, but the Trojans' last-second attempt at a Hail Mary was intercepted.
Troy had only 186 yards of total offense in its 62-7 loss to Mississippi State last week.