Kerr blazed Australia’s first effort over the bar and with Ingrid Hjelmseth then saving from Emily Gielnik it was left to Ingrid Engen to give the Norwegians an unassailable 4-1 advantage in the shoot-out.
It was a dramatic finale to a match that has lit up the tournament, with Norway deserved winners and reaching the last eight for the first time since 2007 although tough on Kerr, who had showed her quality throughout normal and extra time.
England or Cameroon, who play Sunday, await in the quarterfinals, but on the evidence of this match the Norwegians will represent a tough test for future opponents, even without Ballon d’Or winner Ada Hegerberg.
Kerr went close in the opening minute of the match, her shot going narrowly wide, but it was the Norwegians who looked the more dangerous side in the early exchanges.
They were rewarded in the 31st minute as Isabell Herlovsen raced clear to beat Lydia Williams with a well-directed low shot.
Parity looked set to be restored just before halftime as Maria Thorisdottir was adjudged to have handled as she chested a bouncing ball away.
Kerr stood ready to take the penalty but after consulting with the VAR, match referee Riem Hussein reversed her original decision.
Australia came out in the second half with renewed intent and Kerr had the ball in the net, only for it to be ruled out for offside.
The equalizer looked in the offing and came by an unlikely route as a whipped in corner from Elise Kellond-Knight found the net without another player getting a touch.
With seven minutes remaining, Australia kept up the pressure, but in keeping with the end-to-end nature of the match it was the ever dangerous Caroline Graham Hansen who came within a whisker of a late winner.
Her headed touch on a cross hit the post and ran along the line to the relief of the Australian camp.
Into extra time and it was Graham Hansen who brought out a brilliant fingertip save from Williams.
Australia was reduced to 10 just before the end of the first period of extra time as Alanna Kennedy saw red for pulling back Lisa-Marie Utland as she burst clear.
A VAR review could not spare Kennedy and it left Australia holding on with Vilde Boe Risa seeing her effort bounce off the top of the crossbar.
The excellent Graham Hansen set the tone for Norway in the shoot-out by coolly converting with the first of four clinical penalties before Kerr, by cruel contrast, missed the goal entirely with her skied kick.
“Only big players can miss penalties, because small players don’t take them,” said Australian coach Ante Milicic as he reflected on the miss and his side’s exit.
“It was really controlled and a fantastic team effort in the shoot-out,” said Graham Hansen, who was awarded player-of-the-match honors.