Collins fired in a first-half penalty and struck again after Craig Forsyth scored to condemn Barnsley to their first defeat on the road this season.
Derby started strongly with Nathaniel Mendez-Laing causing problems on the left although Joe Wildsmith had to turn behind an Adam Phillips free-kick in the 21st minute.
But Derby went ahead in the 35th minute when Jordan Williams pushed over Collins to concede a penalty which the striker confidently converted.
Derby were worth their lead and they extended it in the 48th minute with Forsyth ghosting in to score with a low finish after a long free-kick was flicked on.
A repeat of that move brought Derby a third in the 63rd minute with Forsyth again getting in on the left to fire in a shot which Ben Killip saved but Collins bundled in the rebound.
Tom Barkhuizen fired over and Louie Sibley rattled the bar as Derby enjoyed a comprehensive victory.