Hernandez put United ahead from the penalty spot and headed in a second goal shortly after half-time before Phil Jones and Fabio da Silva added late goals.
As United coasted into the last eight, it was a night of several positives for manager David Moyes, whose side remain well off the pace in the Premier League.
Experienced defenders Nemanja Vidic and Rio Ferdinand, who had been out since October 5 and September 28 respectively due to injury, both returned to action.
United's unbeaten run was stretched to six matches and Wilfried Zaha, who Moyes has hinted could leave on loan in January, made his first start at Old Trafford since joining from Crystal Palace in the summer.
But it was the performance of another youngster, Adnan Januzaj, that really caught the eye.
The 18-year-old, who signed a new five-year contract earlier this month, has mainly been employed on the left flank, but he excelled in a central attacking role behind Hernandez.
Norwich produced a competitive performance but will be more concerned by news of their Scotland winger Robert Snodgrass, who was carried off on a stretcher after suffering a head injury in the second half.
Januzaj was the dominant influence from the start and, after Nathan Redmond dragged wide for the visitors, the United youngster sent Zaha clear down the right flank and the winger's cross picked out Hernandez, whose effort was deflected wide at the near post.
United were handed the chance to move ahead when Januzaj collected the ball in the area and went down under Leroy Fer's challenge on the byline, with referee Kevin Friend pointing to the spot.
Hernandez made no mistake with the penalty, thumping a shot into Mark Bunn's bottom-right corner to give the hosts the advantage with his third goal in consecutive League Cup ties.
The Mexican almost claimed another when he rose highest to meet Ashley Young's whipped free-kick, but Bunn was well-placed to claim the header.
Ryan Bennett had a header saved by United goalkeeper Anders Lindegaard, making his first start since Moyes took charge, before Young also forced a routine save from Bunn at the other end.
Januzaj went much closer when Jones picked him out in midfield and the youngster drifted forward before bending a shot just past the top corner.
The Canaries threatened after the interval, with Fer missing with a header from a Snodgrass free-kick and Johan Elmander stabbing wide from a Redmond cross.
But United moved into a comfortable position when Hernandez finished off a sweeping passing move early in the second half.
Young picked out left-back Alexander Buttner and the Dutchman clipped over a cross that was met by Hernandez at the near post.
Bunn kept out the striker's first header but could not get a hand to the United man's follow-up as Hernandez squeezed a header past Sebastien Bassong on the line.
Lindegaard was called into action once again when he saved a Snodgrass header at full stretch following a Redmond cross, but the Norwich winger had to be stretchered off after landing awkwardly.
Januzaj went close again when he drifted forward and just missed the target from around 25 yards.
Jones added a third with four minutes left when he volleyed through a crowd of bodies after Bassong could only clear a Januzaj cross as far as the edge of the area.
And United made it 4-0 when Wayne Rooney picked out fellow substitute Fabio with a diagonal ball and the Brazilian prodded in under pressure from Steven Whittaker.
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