The hosts, six points behind Group E leaders Switzerland before the latest round of fixtures, took their time to get going before taking the lead through Elyounoussi's 43rd-minute header.
Goalkeeper Rune Almenning Jarstein pulled off a magnificent save to keep the scores level just after the interval, before King made sure Norway walked away with the spoils in the 66th minute.
The victory temporarily moves Egil Olsen's men up to second in the standings and they would be hoping Slovenia could frustrate Albania at Sportni Park later on Friday night.
Chances were few and far between early on although it was the visitors, with just one win and a draw in their opening six matches, who started brightly with Efstathios Aloneftis forcing Norway keeper Jarstein into action.
The home side steadily started to find their feet and Tom Hogli had the game's best chance midway through the half, when his strike was superbly tipped away by Cyprus goalkeeper Antonis Georgallides, with Brede Hangeland heading wide from the subsequent corner.
Yet Norway failed to capitalise on that, so it came as a bolt out of the blue when they took the lead with the interval approaching as Elyounoussi headed home from Stefan Johansen's corner.
Hangeland was denied by a last-ditch tackle by Giorgos Merkis as Norway started to assert control on proceedings in the second half, while Georgallides did marvellously well to palm away Elyounoussi's effort to keep the deficit down to one goal.
Jason Demetriou nearly levelled for the away side out of nowhere, with his point blank strike forced away by Jarstein.
It proved a crucial save as Norway doubled their advantage soon after as King drilled past Georgallides from Magnus Wolff Eikrem's left-wing cross.