Colombian striker Edixon Perea headed home a cross from midfielder Alejandro Alonso in the 23rd minute and striker Laslandes beat goalkeeper Mickael Landreau from close range with seven minutes left in the first half.
It was the second match within four days between the two clubs after they drew 0-0 in a Ligue 1 match at the weekend.
The visitors, eager to repeat a good performance on the turf of their west coast rivals, reduced the deficit with a penalty by midfielder Mamadou Diall in the 67th minute.
Laslandes, only considered for the bench in the league matches, wrapped it with a stoppage time goal.
Nantes, who eliminated league leaders Olympique Lyon in penalty shootout in the previous round, had come close to equalising when defender Franck Signorino shot wide three minutes from time.
Nice had an easier outing against second division Sedan, the Riviera side booking their ticket for the next round 2-0 thanks to goals by defender Cyril Rool and striker Marama Vahirua.
In a clash between Ligue Two clubs, En Avant Guingamp beat Caen 2-0.
Arnaud Gonzalez put the Brittany side in front on the half-hour and striker Cyril Watier got the second with two minutes to go.
On Wednesday, Paris St Germain are at struggling Toulouse while Monaco host solid Lille, ready to claim another scalp after they handed Lyon their first defeat since the start of the season in last weekend's league action.
