The holders would have secured progress had Northern Ireland failed to defeat Czech Republic, but their Group C rivals' win means the defending champions will advance if they avoid defeat in Belfast next month.
Few would back against Joachim Low's side on this form, after a three-goal blitz in the opening 21 minutes effectively put the game to bed in Stuttgart on Monday.
Mesut Ozil, perhaps relishing the freedom of a break from the current discontent at Arsenal, gave Germany the lead and Julian Draxler - recalled as one of three changes from Friday's 2-1 win in Prague - added a second soon afterwards.
Werner's first was a simple tap-in as Norway wilted under the opening barrage, and the RB Leipzig forward doubled his tally with a fine header shortly before the interval.
Half-time substitute Leon Goretzka made it five shortly after the resumption and his fellow replacement Mario Gomez added further gloss as Germany maintained their perfect qualifying record this campaign.
Germany had taken only quarter of an hour to make the breakthrough in the reverse fixture exactly a year ago and they were even quicker off the mark this time as Ozil swept home Jonas Hector's cut-back from the left in the 10th minute.
Toni Kroos blazed over when he ought to have doubled the lead a minute later but the hosts did not have to wait long to stretch their advantage as Draxler turned sharply inside the box and placed a calm finish into the bottom-left corner.
Norway were in danger of complete capitulation and only had themselves to blame for Germany's third. Jorgen Skjelvik's loose pass set the defence on the back foot and Thomas Muller dummied Kroos' square pass to Werner, who had the simplest of tasks to tap home.
The visitors briefly managed to stem the black and white tide, but the respite was only temporary as Werner claimed his second in the 40th minute, arching his back impressively to head in Muller's excellent right-wing delivery.
Özil's first goal for in over a year— European Qualifiers (@EuroQualifiers) September 4, 2017
Draxler goal on first start of the season
Werner now 6 goals in 8 internationals
Kroos was brilliantly denied by Norway goalkeeper Rune Jarstein on the stroke of half-time but the pattern continued after the break as Muller's replacement Goretzka quickly announced himself on proceedings, rising highest to nod home Draxler's chipped cross.
Norway belatedly showed some signs of life as Ole Selnaes fired narrowly off-target and Mats Hummels deflected a low ball just wide but Germany were soon on the front foot again as Gomez's diving header converted Joshua Kimmich's cross to complete a thumping victory for the holders, whose form looks ominous for those hoping to relieve them of the trophy in Russia.