Sergio Ramos kept Norway at bay until the last-minute leveller ruined a clean sheet as he made history by becoming Spain's most-capped player, passing Iker Casillas with his 168th international appearance.
Saul had given the away side the lead just after the break but Sweden's 4-0 victory over Malta means Spain will have to wait to seal qualification to next summer's finals.
The visitors continued to boss early possession and Jesus Navas tried to set up Rodrigo midway through the first half but his cross across the face of the goal was just out of the former Bolton loanee's reach.
Down at the other end, Norway came close with their first shot on target as Omar Elabdellaoui skilfully sliced a cross behind his left leg to find the head of King in the box, but the Bournemouth striker failed to get any power on the ball and it was safely collected by goalkeeper Kepa Arrizabalaga.
Elabdellaoui had a chance to break the deadlock on the stroke of half-time for Norway but – under pressure from the Spain defence – he fired straight at Kepa.
The visitors came out of the blocks in search of an opener after the break and they were rewarded for their fast start as Saul took advantage of a failed clearance to strike a powerful shot into the back of the net from the edge of the 18-yard box.
King headed just wide as Norway searched for an immediate leveller and 10 minutes later, Saul fired high into the stands as the away side looked to double their score.
Napoli midfielder Fabian struck the crossbar with a long-range attempt to extend Spain's lead after 70 minutes.
King almost netted an equaliser for the home side with 15 minutes remaining, sending a thunderbolt just over the crossbar as Norway pushed for a point and mounted on the pressure.
Real Madrid defender Ramos was booked as the minutes ticked down meaning he will miss the next qualifying match away to Sweden on Tuesday.
Norway got the goal they sought late on as they were awarded a penalty in the fourth minute of added time after Kepa collided with Elabdellaoui in the box.
King stepped up and slotted the resulting spot-kick into the bottom-right corner for a share of the spoils to see fourth-placed Norway fight another day.