Substitute Weimann fired home from close range in the first minute of stoppage time after Rotherham keeper Viktor Johansson palmed out a Kal Naismith shot.
Tommy Conway had put City in front after 15 minutes with a bullet header from Cameron Pring’s cross.
But Rotherham looked to have earned a point when Jordan Hugill sent Max O’Leary the wrong way with a 67th-minute penalty after a foul on him by Zak Vyner.
Robins boss Pearson said: “It has been a difficult season at times, but I think our fans have seen definite signs of progress.
“Now we have the points to back that up. With so many young players having gained experience in the Championship, I believe the right additions in the summer can make us contenders for the top six.
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