The US had taken a two-goal lead in the second period, but Canada were quickly back on level terms, and after the United States went in front again in the third period, Sarah Fillier's goal sent the final to sudden-death overtime.
With less than three minutes remaining in the first period of overtime, Janecke got to the puck ahead of keeper Ann-Renee Desbiens to score from close range before skating away and throwing her stick in the air in celebration.
After a scoreless first period, the second saw four goals in the space of two minutes and 16 seconds, with the U.S. taking a 2-0 lead with goals in quick succession from Caroline Harvey and Abbey Murphy before Canada drew level.
Harvey celebrated the opening goal by miming holding a phone to her ear, after she appeared to lose a mobile phone on the ice during her side's quarter-final win over Germany.
Danielle Serdachny shot through traffic to pull one back, and Jennifer Gardiner scored her sixth goal of her first world championships, but Canada failed to take advantage of two power plays shortly afterwards.
The US, however, made the most of a power play early in the third period, and playing with a 5-on-3 advantage, they went in front again through Taylor Heise.
Canada recovered from the sucker punch, killed off another penalty and Sarah Fillier pounced on a loose puck to equalise with less than six minutes remaining and ultimately send the match to sudden-death overtime.
The US lost first-choice goaltender Aerin Frankel to injury in the third period, but Gwyneth Philips made some vital saves in overtime before Janecke found the winner from Heise's pass in what was the longest game in women's worlds history.
Earlier, Finland won their 15th bronze medal in the competition with a 4-3 overtime win over Czech Republic after the hosts had led 2-0 at the end of the first period.