Storey and Harris were chasing the pace from early on and it took a desperate late surge in the final 250m to get the New Zealanders ahead of Britain and into third place.
That followed the disappointment of single sculler Robbie Manson having his hopes of victory dashed.
The New Zealand sculler had won both World Cup regattas leading up to the worlds and appeared one of the stronger chances to make up for last year's fifth placing.
Instead he wasn't in the contest, finishing fifth again as Norwegian Kjetil Borch dominated the race from the outside lane 6.
The double scull crown went to French pair Hugo Boucheron and Mattieu Androdias, who were in front from early in the race and won in 6min 05.160s.
Switzerland took the silver and Harris and Storey looked to be heading for a fourth place finish.
However they turned on the blowtorch to overtake Britain's Angus Groom and Jack Beaumont as the finish line approached.
In the single scull final, Borch was unchallenged, winning in 6min 38.310s, from defending champion Ondrej Synek of the Czech Republic, who clocked 6:39.920 with Lithuanian Mindaugas Griskonis taking bronze in 6:42.900.
Manson, who has been involved in a battle to hold his single seat from double Olympic champion Mahe Drysdale this year, recorded 6:46.110 and will be bitterly disappointed with the outcome.
By the halfway stage, Manson was 4s off the pace and Borch kept his foot down. Manson had slipped to 5.56s back at the 1500m mark but was up to fourth.
Briefly he looked to have a chance of pushing for bronze, but Griskonis surged up in lane one alongside Manson and finished strongly to take third.