Neil Danns volleyed home his first goal for the club in the 54th minute and although Leeds substitute Aidy White headed home with his first touch soon after, James Vaughan held his nerve to slide home an 86th-minute winner for the visitors, who triumphed 2-1.
It was Robins' third win since he replaced Simon Grayson as boss last month and lifted the club eight points clear of the relegation zone with eight games remaining.
"I thought it was brilliant," said Robins. "It was a typical derby game. The referee set his stall out early doors because he booked a couple of players and I thought there might not be 11 players left on the pitch.
"We grew into the game. We were resolute in the first half, got ourselves in front and conceded a really poor goal from our perspective because we started to drop (deep) and there was no pressure on the ball.
"But we kept our heads and although we were pegged back we were still in the game and although they were throwing bodies forward we dealt with it well.
"Neil's obviously thrown caution to the wind because the crowd demands it here and you think they might leave one or two gaps and we exploited that.
Robins added: "Some derbies can be dire, but I don't think this was anything like that. Some of the quality we showed was good and what I expected because there's some decent players in there."