Second-half goals from Reuben Reid and Ryan Taylor sealed the win and despite a late response from Adam Le Fondre, the Millers held on for the points to move up to 15th in the league.
"I told the players this week I felt they were ready to compete against the best teams in this division and they have done that against a very good Rochdale side," said Robins.
"We put in a really solid, professional and at times clinical performance and we deserved our win.
"I thought everyone was outstanding in very difficult conditions, we mastered those conditions better on the day than Rochdale and in the end we could have won by more goals.
"I was disappointed with the one we conceded, but I'm just being greedy because I wanted a clean sheet," quipped Robins.