The Tigers looked to be facing a humiliating Europa League third qualifying round exit against the Slovakian side after the visitors went ahead at the KC Stadium inside two minutes through Tomas Malek.
But Ahmed Elmohamady levelled for the Premier League side in the 27th minute before Sone Aluko stepped off the bench to score the winner 10 minutes from time and book Bruce’s side a berth in the play-offs for the group stage.
The Hull boss told Sky Sports: “The first thing I said to them was make sure you concentrate.
“We’ve made a mistake and we’ve got off to an awful start and for the first 20 minutes everyone was on edge.
“Thankfully, in the end we stuck with it and got the winner, which I think overall we deserved, we had some great chances and at this level you have to take them.”
Bruce knows Hull will face a much tougher task this season with the added workload of European fixtures, with greater expectation also on their shoulders after surviving in the top flight last term and reaching the FA Cup final.
“The players have got a good spirit amongst them and they are going to need that. We are under no illusions as to how difficult it is,” he added.
“We had a good season last year, so we’ve got to make sure we are ready for the challenge now.
“For me, that was always the biggest challenge - the second season in - because there was no expectation last year at all.
“The players have got the effort and determination to try and succeed.”