The new manager's first home game in charge produced an all-too-familiar outcome as first-half goals from Nouha Dicko and Ben Pringle gave Rotherham a deserved interval lead.
Substitute Bobby Reid replied for the hosts with an 81st-minute goal, but it was too little too late, despite an improved second-half performance.
Cotterill said: "It is never nice to be bottom of the table and I will have to wait to see if it has a psychological effect.
"There is a lot of hard work ahead. We wasted 45 minutes by playing at too slow a tempo and could have no complaints about losing on our first-half display.
"We passed the ball, but passed it too slowly. The second half was a lot better, but it will only mean anything if we start our next game in the same manner.
"I already know what needs doing, but it's quite a lot and will take time. The players need to relax on the ball more and move it quicker if we are to be effective."
Dicko's second-minute goal was his fourth in his last four starts. The striker pounced to score from close range when City goalkeeper Elliott Parish failed to hold a long-range shot from Kieran Agard.
Pringle swept home the second in the 27th minute after a strong run from Lee Frecklington and Rotherham could have been further ahead by the break, having played some slick passing football.
Reid's reply came when his curling cross from the left drifted in at the far post without anyone getting a touch.
Rotherham keeper Adam Collin then had to save well from Sam Baldock as the home side pressed for a late equaliser.