Phil Jones and substitute Fabio also got on the scoresheet in a one-sided encounter played amid frequent and heavy showers at Old Trafford.
Norwich struggled to muster a threat and will be far more concerned about the serious-looking injury suffered by Robert Snodgrass in the second half than their exit.
The former Leeds winger required prolonged treatment after apparently being knocked out in a clash of heads with Rafael and was carried off on a stretcher.
Snodgrass had been in the process of heading at goal when he was injured, one of few chances the much-changed Canaries created.
For the most part Rio Ferdinand and Nemanja Vidic, back together in United's defence for the first time since the derby thrashing by Manchester City in September, were untroubled.
Winger Wilfried Zaha made his first start since the Community Shield but it was Adnan Januzaj and Hernandez who stole the show.
Januzaj, playing just behind Hernandez, was creative and dynamic and the Canaries struggled to contain him.
United had made nine changes, but it was Norwich's that robbed the game of intensity. Snodgrass, Leroy Fer and Sebastien Bassong were their only survivors from the weekend.
Aside from an early run and shot wide from Nathan Redmond, the visitors allowed United to dictate the pace of the game.
Zaha's first serious run at Norwich created a chance for Hernandez but his shot on the turn was deflected wide.
Bradley Johnson made a brave clearance from the resulting corner, taking a blow for his troubles, but the threat was not long repelled.
Referee Kevin Friend pointed to the spot after Januzaj went down on the slippery turf following contact from Fer.
Hernandez lashed home the penalty and went close to a second as he headed at Mark Bunn from an Ashley Young free-kick.
Januzaj curled a good effort narrowly wide from 25 yards and then had two shots blocked in quick succession after a dangerous run into the box.
Hernandez was millimetres away from connecting with a Young cross as United maintained the momentum until the break.
Norwich showed more urgency at the start of the second half and had chances through Bassong, Fer and Johan Elmander but the United goal was not threatened.
United doubled their lead in the 54th minute as Alexander Buttner picked out an unmarked Hernandez in front of goal with a cross from the left.
Bunn parried Hernandez's initial header but the Mexican followed up with another and this time there was no mistake.
Norwich tried to respond and Snodgrass got into a good position to meet a Redmond cross from the right but while his header forced Anders Lindegaard to save, he clashed heads with Rafael in the process.
He immediately hit the turf and Lindegaard was quick to call for medical attention.
Play was stopped for about eight minutes and the Scotland international received warm applause from around the ground when eventually carried off on a stretcher.
United regained their stride and Januazj almost added a third after a good run but his low shot whistled just wide.
Norwich's cause was hardly helped when Moyes decided to introduce Wayne Rooney from the bench, in place of Zaha, after 78 minutes.
But it was Jones who put the result beyond any doubt, volleying back past Bunn after a Bassong headed clearance fell to him on the edge of the area.
Januzaj left the field to a good ovation as the game entered 11 minutes of injury time and his replacement Fabio was quick to make an impression.
The Brazilian took a long ball from Rooney in his stride and calmly slipped it past the onrushing Bunn.