The side were frustrated again on Sunday as Artem Dovbyk's first-half goal proved insufficient to secure their first three points of the season at Genoa as the hosts' Koni De Winter equalised six minutes into stoppage time.
Roma have three points from four matches and are already five points behind leaders Juventus.
"I believe this is a team that has to challenge for the Champions League places. That is our objective," De Rossi told DAZN.
Roma have only gone without a win in their first four Serie A matches in four other seasons before this one.
De Rossi's side have won just one of their last 11 games in all competitions. The manager, visibly frustrated with Sunday's result, was also sent off for dissent just before the equaliser.
"It was the first-half performance we wanted, making Genoa move around and keeping the ball, so we should’ve taken those opportunities to score a second goal," De Rossi said.
"In the second half, we created nothing. The numbers are in our favour but if you dominate in the first half, things shouldn’t be different in the second."
Roma made several signings in the recent transfer window, including Ukrainian striker Dovbyk and Argentine winger Matias Soule, but the wins have yet to come.
"We had a strong transfer campaign to reinforce the squad for our objective," De Rossi said.
"Unfortunately these dropped points weigh heavily on our position in the table - and even more so on our mood."
