Dave Rennie's side were made to work early on and led just 14-10 at halftime after Italy made a bright start through Tommaso Menoncello, before Sam Darry and Jordan crossed before the break to edge the hosts ahead.
The All Blacks emerged with far greater intensity in the second half and quickly broke down Italy's defence, scoring four tries in the space of 15 minutes through Cam Roigard, Ethan de Groot and two quickfire efforts from Jordan to take control of the contest.
Italy's task became even tougher when lock Niccolo Cannone's yellow card for a headbutt on Roigard was upgraded to a red following a bunker review.
Jordan's three-try haul took his career tally to 50 tries in just 56 Tests, moving him past former winger Doug Howlett as New Zealand's outright record try-scorer.
To Italy's credit, they managed to add another try after the red card, Leonardo Marin scoring the visitors' only points of the second half before Tupou Vaa'i added a late seventh try for the All Blacks.
The result gave New Zealand a second successive victory in the inaugural Nations Championship, after last week's win over France. They host Ireland next weekend, while Italy travel to Perth to face Australia.
The Lions gave New Zealand an almighty scare with a dominant first-half performance before the All Blacks were able to fight their way back to a 41-35 win at Ellis Park on Tuesday night.
Lions head coach Ivan van Rooyen admitted his side are a little undercooked for their Rugby's Greatest Rivalry tour match against the All Blacks but believes the occasion will help elevate their performance.
Springboks Pieter-Steph du Toit and Ox Nche both earn career milestones in the second Test of Rugby's Greatest Rivalry against the All Blacks in Cape Town this Saturday.
The Lions have been boosted by the return of Springboks Quan Horn and Boan Venter for their Rugby's Greatest Rivalry clash against the All Blacks at Ellis Park on Tuesday night.
