An opening day League Cup victory at Bury back in August was followed by wins over Gillingham, Preston and North-East neighbours Sunderland to set up a quarter-final tie at Swansea City.
Despite some impressive Swans performances in the Premier League, though, Steele is satisfied Middlesbrough can spring a surprise at the Liberty Stadium.
"It's a big occasion for the players. It is one of my biggest games for Boro and hopefully we can reward the fans and get to Wembley," said Steele.
"All our games in this competition have been away from home this season so that has given us a lot of confidence that we can travel to anywhere and get a result."
It will be the 12th time in a row that Middlesbrough have played away from home in the League Cup, with more than 4,000 miles being racked up by supporters who have taken in them all since the trip to Yeovil in August 2008.
Free coaches - laid on by the club and Capital One - are taking supporters to South Wales from Teesside and Steele said: "It's an amazing run. We have one of the best away support numbers in the league.
"They travel up and down the motorways following us everywhere and they have had a few long trips in this competition already. On a cold night in Swansea that is when you need you fans and hopefully we can give them something to shout about and warm them up. We are within touching distance of Wembley and we have to use that as motivation."
Steele, 22, was only 14 when he watched Middlesbrough end their long wait for honours on the television when they won the competition - then called the Carling Cup - in 2004. And he was not even born the last time Boro went to Swansea and won a cup tie.
That arrived on January 3, 1981 when Middlesbrough - boasting Mowbrays first team coach Mark Proctor in their ranks - won 5-0 at the Vetch Field when Terry Cochranes incredible overhead kick was the pick of the FA Cup goals.