Montella says Turkey proved critics wrong at Euros
Montella's side earned a tense victory over the 10-man Czechs to reach the knockout phase of a major tournament for the first time since a run to the Euro 2008 semi-finals.
Turkey, who have been roared on by vociferous support in Germany, had struggled in a 3-0 loss to Portugal last time out to put their hopes of qualifying from Group F in peril.
"We are always focused and hard-working, we deserve this qualification," said Italian Montella.
"We have suffered some criticism that was not deserved, and we overcame that.
"I'm used to pressure, I don't mind it, but I don't like when pressure is put on my players."
Turkey will next face Austria in Leipzig on Tuesday for a place in the quarter-finals.
The Crescent Stars have made the semis on the last two occasions they reached the knockouts in major tournaments, after finishing third at the 2002 World Cup.
"Now we have new goals," accepted Montella.
"Tonight we just want to celebrate, we deserve it, great sacrifice. But starting tomorrow, we want to continue to make history."
Turkey knew a draw would be enough against the Czechs in Hamburg to go through, but struggled to put the game to bed despite having a man advantage for more than 70 minutes after Antonin Barak was sent off.
Hakan Calhanoglu gave them the lead before Tomas Soucek gave the Czech Republic hope of snatching the win they needed to knock Turkey out.
But Cenk Tosun finally finished off the match with an injury-time winner on the counter-attack.
"We played well, we did not exploit one more man enough," Montella said.
"When we took the lead, we conceded a goal... We've suffered pressure in the last few days and the team is very young.
"We could have suffered a bit less, but it makes our joy even greater."