Austin Peay freshman guard Terry Taylor had 15 points in his collegiate debut to lead all scorers.
After a sluggish offensive start to the second half, the Commodores' 41-21 halftime lead was reduced to 51-42 after Taylor's drive, bank and free throw with 8:51 left.
But Brown hit a layup after a Roberson assist for a three-point play. After Deyshawn Martin answered for Austin Peay with a layup, LaChance canned a 3-pointer from the right corner and Vandy maintained a double-digit lead the rest of the way.
It was a tight first half until Vandy's bench, led by Brown and guards Saben Lee and Payton Willis, broke the game open.
With Vanderbilt clinging to a 23-19 lead, Willis hit two foul shots and then Brown scored a fast-break dunk.
Moments later, Willis helped extend the lead to 12 after back-to-back 3-pointers, one coming off an assist by Lee.
The 6-foot-2 Lee had the biggest highlight of the first half, throwing down a thunderous, one-handed dunk over 6-foot-6 Austin Peay guard Acoydian McCarthy via a Willis steal.
It was the coaching debut for Austin Peay's Matt Figger, who was an assistant at South Carolina last year.
Vanderbilt guard Matthew Fisher-Davis, who led the team with 13.9 points per game last year, sat out with an ankle injury. He's expected back soon.
Roberson hit a 3-pointer for Vandy's first points of the season. It was the 1,000th straight game in which the Commodores have connected from behind the arc. They have hit a 3-pointer in every game since the shot's institution.