Johnson scored from the spot either side of half-time as Wednesday lost their second game of the season.
Wednesday defender Moses Odubajo, who was later hauled off, will want to forget his performance as soon as possible having been responsible for both penalties.
In contrast, Alex Neil's men will now be full of confidence after adding another three points following a resounding 3-1 win over Stoke on Wednesday night.
It is nearly 70 years since Wednesday, who had enjoyed a stellar start to the season, recorded a win at Deepdale in the second tier.
There was not much in the way of chances in the first half, with the Yorkshiremen, who had netted six goals in their previous four games, mustering just two efforts on goal – and neither were on target.
Preston were not much more efficient, hitting the target twice in the first half from two efforts, despite enjoying the lion's share of possession.
They took the lead in the 32nd minute when striker Sean Maguire was brought down in the area by Wednesday right-back Odubajo.
Referee Robert Jones pointed to the spot, with Johnson hitting the ball under Keiren Westwood, although the Wednesday goalkeeper got a hand to it.
Preston should have doubled their advantage minutes later when Maguire sent in a ball to the far post for the awaiting Jayden Stockley, but Wednesday centre-half Julian Borner reached it first and cleared the ball behind for a corner.
The home side made it two in the 65th minute after Odubajo again fouled Maguire in the box and Johnson stepped up to score his second penalty of the afternoon – and his fourth goal of the campaign.
Preston continued to surge forward in search of a third goal, with Johnson and Tom Barkhuizen both going close.
Preston withstood Wednesday pressure in the final 10 minutes as the visitors looked for an equaliser.
Substitute Fernando Forestieri's thunderbolt from outside the box was well saved by Declan Rudd, with the Preston stopper stepping up again to keep out Atdhe Nuhiu's effort as the home side held out for the win.