Gibbs had the upper hand and was serving at 7-5 2-6 2-1 when the players were called off on Sunday evening.
Their match will resume on Monday following the conclusion of Grigor Dimitrov's contest against Viktor Troicki, which was due to start at 10:00 BST on court three, but has been delayed by heavy rain in Paris.
Kyle Edmund was among a host of players scheduled to begin their campaigns on day one but who must now wait on the weather.
Edmund will now be third on court six against qualifier Nikoloz Basilashvili, while fellow Briton Naomi Broady plays Coco Vandeweghe on court four.
British number one Andy Murray is scheduled to face 37-year-old Czech Radek Stepanek in the fourth game on Philippe-Chatrier Court, but, with more rain expected, it is likely they will play on Tuesday.
Australian Nick Kyrgios and two-time Wimbledon champion Petra Kvitova both progressed before the rain delays on Sunday.
Kyrgios overcame a first-set code violation to beat Italy's Marco Cecchinato 7-6 (8-6) 7-6 (8-6) 6-4.
The men's 17th seed picked up the warning for screaming at a ball boy for a towel during the first-set tie-break and then accused umpire Carlos Ramos of "unbelievable bias" for his punishment.
Tenth seed Kvitova needed three sets to get past Danka Kovinic of Montenegro 6-2 4-6 7-5.
Lucie Safarova, last year's losing finalist, breezed through to the second round with a 6-2 6-0 victory over Russia's Vitalia Diatchenko.
Britain's number two Heather Watson fought back against American Nicole Gibbs at the French Open before play was suspended because of rain.