The Ayrshire club are languishing in the bottom two of the Scottish Premiership following a string of poor results, including last week's defeat to Motherwell.
The cup competition is set to provide some respite from league business and Johnston believes a win at Tannadice could be the boost Killie need for forthcoming matches.
The midfielder said: "We need to try to win this game and stay in the cup.
"If we get a good result, hopefully we can take that into next Saturday's game in the league and push on from there.
"We are still disappointed after last week's game but we know things can change with a couple of wins.
"Hopefully we can win this one. With the cup, you never know what can happen.
"Facing Dundee United will be a hard game but we know if we play well we can get a win."
Johnston is well aware that facing United away from home is a particularly tough fourth-round tie but is confident about Killie's chances.
He added: "They are flying and we know all about their attacking threats.
"If we can keep them quiet at the back, we know we can go forward and attack ourselves.
"They deserve all the credit they are getting, especially the likes of Ryan Gauld, Andy Robertson and John Souttar, they are doing great.
"They are a great attacking side but we can get a good result.
"On our day, we can cause folk problems. We just need a break.
"We know if we play as well as we can do, we can beat Dundee United."
United manager Jackie McNamara has called on his side to continue their fine run of form in the cup clash.
The Tangerines are enjoying a purple patch which saw them move up into fourth place in the Premiership thanks to last weekend's 4-1 victory over Partick Thistle.
And McNamara has challenged his players to reproduce the form they have been showing in recent weeks as they bid to progress in the cup.
He said: "It's a big game. We'll approach it as we do every one by trying to win it.
"Our form recently has been good, which is something we are going to try to continue.
"It's about what we do and we always focus on ourselves.
"This game will be no different, especially with what is at stake for the club.
"It's one we're looking forward to and hopefully it's a positive night for us."