The England striker netted from two long balls to get United within sight of victory against their bitter rivals.
But they were flying in the first half thanks to two long punts forward from David de Gea – and Romelu Lukaku winning headers.
Lukaku flicked in for Rashford to speed into the area and open the scoring with a powerful finish.
He doubled the lead when Lukaku won another duel with Dejan Lovren.
They pulled one back when Bailly sliced into his own net from Sadio Mane’s cross.
And they could have had another spot-kick when Marouane Fellaini blocked Mane in the area.