The visitors made a fast start at the Via del Mare, with Osmajic opening the scoring in the 11th minute before Messias doubled their advantage in the 29th. Youssef Maleh pulled one back for Lecce from a deflected free-kick, but Nikola Stulic's apparent equaliser was ruled out after a VAR review for handball.
De Rossi was delighted with the response from his players, particularly after fearing they might approach the match with too much confidence.
"We are very happy. The team at the top of the table is strong, and today I was afraid we might arrive here feeling a little complacent, but we did well in the first half-hour. Then we made some mistakes in possession and could have finished the match."
The Genoa coach said his squad had retained its hunger despite climbing away from the bottom of the table.
"We still have a great deal of hunger. We are level on points with teams that, until recently, seemed a long way away from us. I have worked on the mentality of the players and tactically, but I had done that in the first five matches as well."
He was also keen to credit the supporters for their role in the team's recovery.
"The fans, we thank them because there were so many of them. Our stadium lit the spark. The players are the ones who pull you out of the quicksand. I always had the feeling that there were people who believed in us."
Lecce coach Eusebio Di Francesco felt his side had paid for mistakes at crucial moments and a lack of quality in the final third.
"If you win the ball high up the pitch and don't make the most of the first opportunity, they score. They were better than us in their final decisions, apart from our mistakes. There were moments when we gave the ball back to them too easily, and they made the most of it."
He was frustrated by the contrast between Lecce's attacking effort and the outcome.
"In the second half the team was always there, pressing, but we lacked quality in the final decisions. It is difficult for me to analyse those situations."
Di Francesco also questioned the consistency of the refereeing, particularly over the penalty decisions.
"I don't understand anymore what is a foul. I have to tell my defenders to bring attackers down because apparently nothing gets given. Today I saw unusual things, starting with the penalty awarded to Frosinone and the one not given to us."
The Lecce manager accepted that his side needed to improve defensively, rather than search for excuses.
"We have to reverse the trend. We are conceding goals too easily. The performances are there, but the results are missing. I want to take responsibility for making myself understood better. We have to fight for survival until the end."
Genoa's third consecutive victory continued their climb, while Lecce suffered a third successive league defeat.