The Rome club, hosting the second leg of the quarter-final behind closed doors due to recent racist behaviour by their fans, faced an uphill battle having lost 2-0 in Istanbul last week.
For 70 minutes they put up a determined fight and Senad Lulic's goal just after the hour mark gave them hope, but Caner Erkin's powerful equaliser - the visitors' first shot on target - ended Italian hopes and put Fener into the final four of a European competition for the first time.
"We compromised our qualification chances in the first leg," Petkovic said on Sky Italia.
"However, congratulations to the players. I'm happy with how they performed in the game - commitment right up to the 90th minute.
"We tried to win it, even if it no longer counted. We saw a Lazio who produced 15 times what the opponents did (attempts on goal). With a little more precision we could have won.
"Now we need patience and to rest well. We will take a breather before a good game on Monday evening against Juventus and will do everything to win."
Germany striker Miroslav Klose was introduced in the second half and failed to have the desired impact, but Petkovic refused to point the finger at any individual.
"I cannot blame anyone," he said. "Everyone gave their all - they fought right to the end."
Petkovic admitted the team missed having the support of their fans as they battled to continue their European campaign in the eerie surrounds of a virtually empty Stadio Olimpico.
"They would have helped," he said. "I'm sure, especially at 1-0 our fans would have given us the edge."
He added on www.gazzetta.it: "One of the regrets is not being at 100 per cent in this tie, but another is that we had 15 shots on goal and did not win this game.
"But we should not cry over things that cannot be changed. The boys gave their all to turn around the negative result of Istanbul, but I'm very happy."
Fenerbahce forward Dirk Kuyt hailed a momentous day for the Turkish club.
Quoted on spor.haberturk.com, the Holland international said: "Today [Thursday] was an historic day indeed. Rome is an historic city and although we arrived leading 2-0 it is not easy to maintain.
"We worked hard and did not panic when we went a goal down.
"Reaching the semi-finals is something for all of us, and is very important for all of Turkey."