Slovakia happy to secure qualification but braced for tough last-16 game
The Slovakians could next face group winners Spain or England.
"We have world-class opponents ahead," he told reporters. "Regardless of opponents it will be hard."
Slovakia midfielder Stanislav Lobotka, player of the match against Romania, agreed.
"It will be very difficult, but we will try to knock them out," he said.
Slovakia took the lead against Romania in the first half when Ondrej Duda scored with a header, but Romania equalised from the penalty spot through Razvan Marin before the break.
"To be honest, I feel exhausted," Lobotka said. "But whenever you have success you draw energy from it, from the fans, from the stadium."
The tough weather conditions had made the game even more difficult, he said.
"You have that heat and hard ground in the beginning, after 40 minutes it changes completely," Lobotka said of the driving rain and thunderstorms.
Group E presented an unprecedented scenario, with all four teams tied on three points heading into the final day, with a draw sufficient for both Slovakia and Romania to advance.
But Calzona said his instructions were clear from the beginning - win the match.
"I am happy with the progression of the last 16, it was important," the Italian said.
Slovakia are appearing in their third straight European Championship. They reached the round of 16 in 2016 and exited at the group stage in the last edition of the tournament.