Jones and Dawson had to better their initial runs to make it through to the semis.
Dawson, the 28-year-old London Olympian, was returning to Europe after winning the Malabar River Festival Extreme Race crown in India last week.
His first run was rusty, and he clocked 96.46s with a penalty touch. He stormed back in his second run with a clean 91.28s effort. That set him up for his second final of the season before the penalty that put him out.
Jones, meanwhile, picked up a touch on the third gate of her first run to finish one spot out of the top-20 automatic qualifiers, clocking up a time of 105.39s.
The 24-year-old double Olympian was solid in her second run, however, finishing fifth to comfortably make it through to the semifinals.
Tauranga's Callum Gilbert, who made the semifinals of the previous World Cup event in Slovakia, picked up five touches in his first round to lie 60th, but had a clean second run to finish just 1.34s away from repeating the feat.
New Zealand's leading whitewater kayakers have one more chance to iron out wrinkles this week, before the countdown to Olympic qualification begins.
After the French World Cup, the paddlers have four weeks to prepare for the canoe slalom World Championships in London next month, where they need to finish in the top-15 to qualify for the Rio Olympics.
New Zealand Olympic team Chef de Mission Rob Waddell says the one-year-to-go milestone is crucial.
"It's when the blowtorch really comes on," said the gold medal winning Olympic rower.
"Suddenly everyone is talking about the Olympic Games and it's a time when New Zealand athletes will be asking themselves some pretty hard questions."
Planning and preparations for the team are well progressed, and strong results from international world cup and world championship results point to a record number of athletes, potentially more than 200, being eligible for team selection.
