Breckin took charge of the Millers for their 3-1 home win over Chesterfield in League Two on Saturday and did his chances of succeeding Robins no harm after goals from Danny Harrison, Adam Le Fondre and Kevin Ellison had cancelled out James Lowry's 21st-minute penalty.
Breckin said: "The first half was a typical derby with both teams showing each other too much respect.
"At half-time, I told the lads to be positive and the second-half performance was magnificent.
"The goals were magnificent as far as I'm concerned. The second half we put them on the back foot, especially when the goals were going in."
The Don Valley result took Rotherham top of the table, above Bournemouth on goal difference.