Italy produced a dominant showing in Athens, where a 10-minute, three-goal burst in the first half wrapped up a third victory from as many Group J matches.
Nicolo Barella, Lorenzo Insigne and Leonardo Bonucci were all on target, while the Azzurri also maintained their perfect defensive record.
However, Mancini's assessment of the display was hardly glowing.
"I don't think it was our best performance," the head coach told Rai Sport. "The important thing is that we continued to improve, we did well from the first minute and put the match on the right track.
"The most important aspect now is: why didn't we score a fourth goal in the second half? We should've pushed harder, but at times we sat back and should've converted some of the other chances.
"We risked conceding, as Greece nearly hurt us on the counter a couple of times.
"I said we're ahead of schedule, but still have a lot of work to do. The Nations League didn't go that badly either, as we were in the running until the last game."
Insigne's goal, a neat finish after skipping past Andreas Samaris on the break, was his first for his country in more than a year, leaving the Napoli stalwart relieved.
"I needed a goal, as scoring with this jersey is special," he told Rai Sport, with Bosnia-Herzegovina up next.
"Above all, we needed the three points and now we have to recover our energy, as there is another battle on Tuesday.
"I think winning that game can be a big step forward towards the Euros, but it'll be intense, and they need points. If we play the way we did [against Greece], we'll achieve our objective.
"It's not just me, as many other players are all working together with the right intensity both in a match situation and in training. The coach is drilling that mentality into us.
"I am happy with the goal, but we're a good group here and if we are to go far, we need everyone."