Cueto scored after 15, 22 and 26 minutes inside Dunedin's covered stadium to ignite England's third consecutive win but first really convincing performance.
The strapping winger was making his first appearance of the tournament after recovering from a back injury.
He combined beautifully with fellow outside backs Ashton and Ben Foden as England ran the Romanians ragged.
Ashton scored twice in the first half and completed his hat trick with 10 minutes remaining.
It was 34-3 at halftime and the English did not let up, scoring again through snappy halfback Ben Youngs just 20 seconds into the second spell.
Within minutes, young centre Manu Tuilagi broke through the line at pace and fed fullback Foden for a simple try.
It was, essentially, English rugby but not as we know it, and it thrilled the crowd of 25,687.
It was all Tuilagi midway through the half when he received the ball from a quick tap and simply bulldozed his way through two defenders.
Flanker Tom Croft and Ashton then sealed one of the more impressive wins at the World Cup.
England showed their intent right from the start, with Youngs delivering quick ball and imploring his backline to use it.
A couple of bullocking runs from Tuilagi appeared to give the Romanians the jitters, and they treated defence as an optional extra for the rest of the first half.
Simple, quick ball led to all three of Cueto's tries being scored on the left.
Ashton showed lovely vision to hit the line and grab England's fourth try, and he pulled out his extravagant swan dive when he scored the fifth.
Romania could only respond with a Marin Dumbrava penalty, though Iulian Dimutras came close with an attempt from 51m.
England 67 (Mark Cueto 3, Chris Ashton 3, Ben Youngs, Ben Foden, Manu Tuilagi, Tom Croft tries; Jonny Wilkinson 3 cons, pen; Toby Flood 4 cons) Romania 3 (Marin Dumbrava pen).
HT: 34-3