Coach Eddie Jones had urged his team to "light up Rome" on Sunday and, in response, they delivered a five-try bonus-point rout orchestrated by man-of-the-match Smith.
Smith cruised over for the opening try in the 10th minute before hooker Jamie George crossed from close range twice before halftime, with England 21-0 ahead at the interval.
Elliot Daly ran onto a beautifully flighted mis-pass from Smith to go over unchallenged in the left corner five minutes into the second half and replacement prop Kyle Sinckler trundled in for the fifth in the 73rd.
Smith converted four of the tries to finish with a 13-point haul.
The win was England's 29th in a row against Italy — and more embarrassingly for the Azzurri, a 34th straight defeat in the Six Nations.
England bounced back successfully — as widely expected — from their 20-17 loss in Scotland in the opening round while it's two defeats on the spin for the Italians, who haven't won a game in the tournament since February 2015.
Italy haven't got within 21 points of England in any of their past 10 meetings.
"It was a good performance," England coach Eddie Jones said.
"We started the game well, got a bit of a flow, started the second half well, and defended well at the end...
We are happy with that ... they (Italy) have played New Zealand and France and in the first halves been in the game – and they were not in the game today."
Ben Youngs came on as a second-half replacement to equal Jason Leonard's record as England's most capped player on 114 appearances, with visit of Wales to Twickenham the likely setting for him to surpass the milestone.
Italy traditionally provide stiff opposition for an hour until eventually being worn down, but it didn't take long for England to get off the mark when Max Malins and Smith combined down the left for the opening try.
Big carries from Nick Isiekwe and Maro Itoje shoved Italy's pack backward and, as the pressure built, George burrowed over from close range.
The first half finished with George using his strength to run through a tackler and reach out to claim his second try but the damage was done by slick handling and a piercing run from Freddie Steward.
It began to look ominous for Italy as Randall weaved a path into space and the hosts' scrum conceded a penalty.
When England opted for the set-piece again, Smith flashed his classy long pass for Daly to score the headline try of the match.
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