Johansson’s save proved the difference at the end of a terrific clash against the gutsy Valiants.
A Millers side which featured seven changes from the line-up which started Saturday’s league stalemate at Oxford opened the game encouragingly.
They almost went in front in the 16th minute at the end of an eye-catching move down the Vale left.
The culmination saw Ollie Rathbone whip in a terrific cross for Will Grigg, who swivelled before lashing in a shot which was well saved by Aidan Stone.
The visitors then survived strong appeals for a penalty after Grigg took a tumble in the box after he darted in to meet Kieran Sadlier’s cross.
League Two high-fliers Vale weren’t overawed, however, and they went close when Nathan Smith scooped an effort over the top from close range.
Shortly before the interval the Millers almost broke the deadlock when Rathbone’s fired in a shot which was deflected narrowly off target.
Millers boss Paul Warne made a couple of substitutions at the break, but it was Vale who created the first chance of the second period.
The impressive Dennis Politic was denied by a smart save from Johansson.
The Millers - now unbeaten in 17 games in all competitions - made the breakthrough in the 56th minute. Sadlier crossed in for Michael Smith, and he slotted home his 14th goal of the season from close in.
Vale sub Amoo twice went close in the closing stages, before he did set up a penalty shoot-out with just two minutes left when he pounced on a rebound.
Rotherham United: Johansson, Harding, Ihiekwe (Edmonds-Green, 64), Mattock, Sadlier, Odoffin, Rathbone (Miller, 64), Lindsay, Bola (Ferguson, 46), Kayode (Ladapo, 76), Grigg (Smith, 46). Unused subs: Chapman, Hull.
Port Vale: Stone, Jones (Benning, 59), Walker, Legge (Amos, 79), Smith, Garrity (Burgess, 59), Johnson (Martin, 59), Conlon, Politic (Amoo, 59), Rodney, Cass. Unused: Covolan, Jones.
Referee: P Wright (Merseyside).