Mikel Arteta's side took the lead midway through the first half when Nketiah deftly redirected the rebound from a Kieran Tierney shot that came crashing off the far post past Bodo/Glimt keeper Nikita Haykin.
And Arsenal were two up shortly after as Holding nodded past Haykin at his near post from an expertly whipped in cross from Vieira that stood after a brief VAR check for a foul in the buildup.
Chasing a two-goal deficit following the break, Bodo/Glimt had more of the game in the second half, creating half-chances with Amahl Pellegrino seeing his low shot superbly saved by keeper Matt Turner in the 57th minute.
However, they still rarely threatened Arsenal with the Premier League leaders adding a third before full-time when substitute Gabriel Jesus tip-toed down the touchline and set up Viera to smash home from close range.
After the match, Arteta praised the 23-year-old Nketiah, who has two goals and an assist in 10 games in all competitions for Arsenal this season.
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"Obviously his work rate, attitude and application is just phenomenal, but he's got this natural ability [to score] and he does it every time," Arteta told reporters.
"I am really happy to have him in the team. He has the qualities to play for this team in any role -- as a starter, finisher, however you call him.
"You have to look at what you have done [which is] great, but what can you do better? That's his mentality and this is where we want to take him. When you see how he has evolved in the last year or two, he still has a lot of room to do that."
The two teams will meet again in Norway next Thursday for the reverse fixture, while Arsenal host Liverpool on Sunday in a savory top-flight match in England.
Arsenal lead Group A with six points from two matches, two more than third-placed Bodo/Glimt, who have played an extra game.