Germany, ranked number two in the world, started strongly and took the lead in the sixth minute when striker Anja Mittag knocked home the ball from close range after Norwegian goalkeeper Ingrid Hjelmseth spilled a routine shot.
The goal was Mittag's fourth of the tournament and followed the hat-trick she scored in the 10-0 thrashing of Ivory Coast in the opening game on Sunday, a result that marked Germany as one of the favourites.
Germany dominated the first half and looked sure to score more goals against a very ponderous Norwegian side.
But Mjelde secured the draw in the 61st minute with a free kick from 20 yards out that kissed the underside of the bar as it went in and the Norwegians finished the game strongly.
Germany coach Silvia Neid lamented her side's inability to convert their chances and said too many passes had gone astray in the second half.
"We lost track of our plan and when you're up against strong opponents the score can very quickly become 1-1, and that's what happened," she told reporters.
The result leaves Germany and Norway tied on four points at the top of the group after two games and the teams -- who are both past winners of the World Cup -- are virtually certain to qualify for the second round.
Mjelde's goal provided some redemption for the midfielder, who missed a penalty during the opening 4-0 win against Thailand on Sunday.
The eventual result had looked very unlikely after a first half where a dominant and fleet-footed Germany outshot the Norwegians 6-1.
"We were lucky. They could have killed the game in the first half," said Norway coach Even Pellerud, who credited the turn-around to a tough team talk at halftime.
The European sides are old rivals and met in 2013 in the final of the European Championships, which the Germans won 1-0 after a goal by Mittag.
Germany claimed the World Cup in 2003 and 2007 but crashed out in the quarter-finals in 2011, when they hosted the tournament. Norway won the cup in 1995.
Outsiders Ivory Coast and Thailand play each other later on Thursday in Ottawa.
(Editing by Mark Lamport-Stokes)
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