The top seed, chasing an elusive first title in Paris and a career Grand Slam, came onto Court Philippe Chatrier with light rain still falling.
Only 37 minutes of play was possible but in that time, the Serb star was broken three times while carving out two breaks of his own.
The 29-year-old will cross the USD 100 million career prize money barrier if he makes a 10th Roland Garros quarter-final.
But having lost the first set 6-3, he would have been relieved to leave behind the ghostly show court where just a smattering of spectators, huddled beneath umbrellas, braved the miserable chill.
Djokovic had a 4-0 career edge over 14th seed Bautista Agut going into Tuesday's match and won their last clash on clay in Madrid for the loss of just three games.
He had even broken serve in the first game Tuesday after an opening point played out over 30 strokes.
The heavy conditions were also playing into the hands of the underdogs elsewhere on Tuesday in last-16 ties.
On Court Suzanne Lenglen, Tsvetana Pironkova of Bulgaria reeled off all six games played to overturn a 6-2, 3-0 deficit against second seed Agnieszka Radwanska in a match suspended on Sunday night.
On Court One, sixth-seeded Simona Halep was leading Samantha Stosur 5-3 on Sunday but the Australian roared back Tuesday to take the first set 7-6 (7/0).
Stosur was 3-2 up with a break when play was halted.
Later Tuesday, if conditions allow, defending women's
champion Serena Williams will play Elina Svitolina of Ukraine.
Andy Murray, the second seed, was due to play his quarter-final against Richard Gasquet on Chatrier although that was likely to be put back until Wednesday.
French Open organisers were also looking at taking another financial hit as rain continued to swamp Paris.
However, with weather conditions set to improve from Wednesday, Forget was reluctant to suggest that the men's final would be pushed back to the third Monday.
"That's looking at the really dark scenario. We're not that bad yet," he said.
"But eventually if they have to play two matches in a row, they will do it. If they have to, I guess they will do it, as well."
The men's final has only been played on the third Monday on two occasions -- in 1973 and 2012 when Rafael Nadal defeated Djokovic.
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