Saul Niguez's goal 62 seconds into the second half looked to have extended Spain's perfect record to seven wins from seven in Group F at Ullevaal Stadion on Saturday.
But Kepa Arrizabalaga's foul on Omar Elabdellaoui in stoppage time handed Norway a golden chance to deny Robert Moreno's men the points they needed.
King stepped up to beat the Chelsea goalkeeper from the spot on the day Spain captain Sergio Ramos broke Iker Casillas' appearance record for La Roja.
Sergio Ramos becomes Spain's most-capped player ever— UEFA EURO 2020 (@UEFAEURO) October 12, 2019
World Cup 2010
EURO 2012 pic.twitter.com/o0f0APw2Bs
Spain, who won the reverse game 2-1 in March thanks to a Ramos penalty, dominated possession from the off but Norway created the first half's clearest chance in the 22nd minute.
Elabdellaoui delivered a rabona cross from the right but King could only head straight at Kepa.
Spain struck a seemingly decisive blow when Rune Jarstein's error saw his side fall behind.
Norway repeatedly failed to clear their lines moments after the restart and when Saul hit a bouncing ball from the edge of the penalty box, it flew through Jarstein's hands.
Fabian Ruiz twice went close to adding a second either side of King blazing over the crossbar, a missed chance that looked to cost Norway a draw.
But Kepa's error, missing the ball with an attempted punch and connecting with Elabdellaoui, allowed King to score from the spot and denied Ramos the perfect way to mark his milestone.
What does it mean? Spain made to wait
Since replacing Luis Enrique as Spain coach in March, initially taking over on a temporary basis, Moreno had seen his side win all their games. But this was arguably La Roja's worst display with Moreno in charge and they failed to score a second goal to secure all three points.
Saul almost decisive for Spain
In a game of few chances, Saul's opening goal should have been key. While the goalkeeper ought to have kept it out, Saul's dipping strike was too good for Jarstein. The Atletico Madrid star could not have scored his first goal of the campaign at a better moment and Kepa's late gaffe prevented him being the matchwinner.
King penalty spares Jarstein
Norway's Euro 2020 hopes looked set for a major blow due to Jarstein's error. He was lucky another goalkeeping mistake at the other end allowed King to score from the spot, as he did against Spain in March. Norway now have their fate largely in their own hands, playing the Faroe Islands and Malta next month.
Spain will look to secure Euro 2020 qualification on Tuesday, when they are in Scandinavia again to take on Sweden, where they will be without the suspended Ramos. Norway will hope to maintain momentum after a sixth qualifier without defeat on the same day, when Lars Lagerback's side are away to Group F's third-placed side Romania.