The Giallorossi fell steadily down the table through the winter under previous boss Zdenek Zeman, with a frustrated board sacking the Czech as they postponed plans to compete in Europe for another year.
Andreazzoli has supervised back-to-back wins for the first time since December, with Vasilis Torosidis settling a 3-2 victory at Atalanta to build on a fine win over Juventus, but Roma are still seven points shy of third place and their city rivals Lazio.
Andreazzoli told newspaper La Repubblica: "The Champions League? We have certainly regrouped but it is pointless to set ourselves targets.
"We just need to focus on giving our all every single Sunday. Our objective is to keep on improving. If we can do that then we will end the season satisfied with ourselves.
"I'm happy we followed up the win over Juventus and it's important that we carry on down this path. Sure, I will readily admit that time is our enemy."
Defender Leandro Castan and striker Mattia Destro remain sidelined through injury as Roma look to seal a seventh straight home win over Genoa, who have not earned a point from this fixture since January 1994.
Genoa boss Davide Ballardini has a very different battle on his hands as the Rossoblu fight to secure their Serie A status, and the former Lazio coach was disappointed his players felt they could settle for a single point from last week's goalless draw with bottom side Palermo.
"We could have done more and in certain situations we could have got forward with the ball at our feet but didn't manage to do so," he said.
"We were not exactly brilliant and I have to give credit to Palermo, who gave everything out on the pitch. For them getting the point was a matter of life and death and it's not good that we just settled for it.
"That should not happen as we have the quality to play much better. Getting all holed up in our own half is not something that we want to happen."
Genoa continue to secure positive results despite the absence of a host of injured regulars including Bosko Jankovic, Antonio Floro Flores, Ruben Olivera and Thomas Manfredini, while Andreas Granqvist and Matuzalem serve one-match bans against Roma.