Despite being an original MLS team, the Red Bulls have never won a U.S. Open Cup, the world’s third-oldest Cup competition.

But a win in Tuesday’s night’s 8 p.m. semifinal — their first since 2003 — will leave them one step from changing all that.

They’ll face FC Cincinnati of the United Soccer League before what is expected to be a raucous sold-out crowd of more than 30,000 at The University of Cincinnati’s Nippert Stadium. And despite facing their second game in three days, a stingy defense and the tourney’s Cinderella story, the Red Bulls are tantalizingly close to a first championship and know they shouldn’t stumble so close to the finish.

“It’s the furthest we’ve been, and no disrespect to Cincinnati — we have to turn up in that game and be ready for the atmosphere — but we should win that game,’’ striker Bradley Wright-Phillips said. “Don’t mistake this for being cocky, but with the quality we’ve got, if we lose this it’ll be on us. We would be the only people to blame for that.”

Perhaps not; but the Red Bulls have suffered plenty of humbling losses in the U.S. Open Cup. They played four semifinal between 1997 and 2003, and only reached one final, shut out 1-0 by Chicago in ’03. They are expected to use a full-strength lineup Tuesday to try to return to the title game.

“Everyone around here is really excited for Tuesday, really excited for a big game against a club and a team we don’t really know,’’ coach Jesse Marsch said. “And it’s going to be in a stadium that there’s going to be more energy in that stadium than maybe we felt all year, and a team that’s playing in the most important game in many of their lives.

“To be fair, it’s an important game in many of our guys’s lives, so it sets up to be a great match and a fun match and one that we’re going to take very seriously and we know is very important.”

Three-time champion Sporting KC will host the final Sept. 20, meaning the Red Bulls will need to win back-to-back road games to become champions. That’s something no team has done since the tournament moved to home venues in the semifinals and finals in 2000. And Cincinnati will be a daunting home venue.

The crowd sold out in just seven hours, ravenous in support of its young team, only in its second year of existence. But it hasn’t taken them long to write a storybook run, the first USL side since 2011 to make the U.S. Open Cup semifinals. And they did it with stifling defense.

Cincinnati has won five games in the tournament, four by a 1-0 score, and beat Chicago in PKs after 120 minutes of scoreless soccer. They have not allowed a goal in nearly eight hours of soccer, but will be without forward Djiby Fall, who has scored all four of their goals.

“It’s a huge game for this organization. It’s a huge game for this group of guys,’’ Red Bulls keeper Luis Robles said. “We know, with a sold-out crowd, they’re going to be up for it. So we’re not taking this game lightly and we expect the best available eleven from them and we expect them to really bring the game to us.”

The Red Bulls have gotten goals from Wright-Phillips, Sacha Kljestan and Daniel Royer throughout the tournament. Royer, however, is out with a knee injury. They are limited to five foreign players, but Wright-Phillips and Felipe qualify as domestics because they are green-card holders.