Italy required victory by at least two goals to progress following their early draw against Brazil and defeat against Korea Republic.
Nigeria, on the other hand, went into the match holding a one-point advantage at the top of Group B and knowing victory would assure progression as group winners.
Ordega doubled the advantage just prior to the interval, before completing her treble just two minutes into the second stanza.
The second goal was perhaps the most notable, with Ordega finishing a neat counter-attack after Esther Sunday played the final pass.
Substitute Osarenoma Igbinovia iced the cake with a fourth a few minutes from full time as Nigeria booked a quarter-final berth.
"Nothing has changed between 2010 and 2012, except that all the girls who were already there have worked hard and improved," said Nigeria coach Okon Edwin.
"We always come to the World Cup to win it. We were close in 2010. We still aim for the final, and this time, we just hope that the result will be different."