The four-time world champions have put the failure to qualify for last year's World Cup behind them, and have now been transformed under new coach Roberto Mancini.
Nicolo Barella opened the scoring after 23 minutes with Lorenzo Insigne and Leonardo Bonucci adding two more in a 10-minute blitz which followed.
Mancini's side have now scored a total of 11 goals and none conceded in their three Group J qualifiers against Greece, Finland and Liechtenstein.
The Italians next play Bosnia-Herzegovina, who fell 2-0 to Finland, in Turin on Tuesday, while Greece play Armenia, who beat Liechtenstein 3-0.
"No, it wasn't my best Italy," said Mancini, who took over in May 2018.
"For example, in the second half we had to score more goals. This needs to be improved.
"After a year of work, we are ahead of schedule, we improve match after match, but there is still a lot of work to do."
Italy have nine points, with Finland second with six points, followed by Greece and Bosnia-Herzegovina on four.
Andrea Belotti marked his return to the Azzurri by setting up 22-year-old Barella to rifle in just his second international goal.
Insigne then broke through after half an hour curling in from 15 metres out to leave Vasilios Barkas with no chance in the Greek goal.
Three minutes later the Greeks were reeling when Bonucci latched onto an Emerson Palmieri cross to head home.
Italy could have scored more with Barkas denying Insigne and Alessandro Florenzi late, and Fiorentina star Federico Chiesa sending over when alone in front of goal.
"The most important thing now is to understand why didn't we score a fourth goal in the second half?" said Mancini.
"We should have pushed harder, we risked conceding, as Greece nearly hurt us on the counter a couple of times.
"There's still a lot of work to do but the important thing is that we continued to improve and had the right attitude."