Stanley were drawn against the very same opponents earlier this month and produced one of the round's stand-out results as James Beattie clinched his first-ever managerial win thanks to a 2-1 success.
The confidence gained by such a victory has yet to transfer to Accrington's league form given they are still winless, but Hunt envisages a cup run on the horizon should they dump out the Premier League Bluebirds on Wednesday.
Speaking to Press Association Sport, the 29-year old said: "When you get good results away at Middlesbrough, a top championship team, it always spurs you on to want to do well in the cup.
"It gives you an extra bit of belief to get to the next round and even the quarter-final. I think that's a real possibility if we get past Cardiff."
Such lofty goals would have been commonplace for Hunt during his stint with Wanderers, for whom he played in Europe and made his Premier League debut against a Manchester United squad including a certain Cristiano Ronaldo in their ranks for the first time.
And having faced lower-league opposition with a top-flight team before, he anticipates Malky Mackay's team will be thoroughly professional.
"Nowadays you have to give every team the respect they deserve," Hunt added.
"When I was at Bolton we played teams such as Walsall, Tranmere and we never once went into that game thinking this is going to be an easy ride here, we'll just turn up and win.
"We always gave them the respect. We always knew their positives but we also knew their weaknesses as well, and we'll just try to exploit them and I think that's what we'll try to do against Cardiff."
Hunt, who along with the suspended former Manchester United forward Danny Webber will be sidelined on Wednesday, revealed that the entire Stanley squad was gathered around Beattie's phone watching the draw live when they were pitted against Cardiff.
Captain Luke Joyce said: "Wednesday night is a bit of a bonus for us against a Premier League team. It's just going to be fantastic for the club, the players, the fans.
"I think some of the Scouse lads were hoping for Liverpool or Everton but it's a fantastic draw to get Cardiff at home.
"I don't know if the teams have ever played each other before but for us to test ourselves against Premier League players...it's going to be a great night."
Stanley may be without Hunt and Webber but Ian Dunbavin was in goal for Shrewsbury in 2003 when they famously ousted Wayne Rooney's Everton in the FA Cup.
- Sports & Recreation
- Premier League
- James Beattie