Wawrinka, who is ranked No. 3 in the world, received a first-round bye, then let a match-point opportunity slip away in a 3-6, 7-6 (6), 7-6 (1) loss to qualifier Jan-Lennard Struff of Germany.
Wawrinka was hoping to bounce back after losing last week in the quarterfinals of Basel to No. 53-ranked Mischa Zverev. But instead he lost to Struff, who is ranked 91st.
Wawrinka won the first set and had a match point at 6-5 in the second-set tiebreaker. But Struff won three straight points to claim that set and won the deciding set in a tiebreaker as well.
Murray, who is hoping to take over the No. 1 ranking this week, was pressed in his opening match but beat Fernando Verdasco 6-3, 6-7 (5), 7-5.
"It wasn't the best match from my side. I got through it. I will play better tomorrow," Murray said. "I didn't necessarily feel comfortable out there, but I feel like the way the two of us finished the match, I was stronger at the end in the last game when I had my chance."
Murray saved two break point chances at 5-5 in the third set, then broke Verdasco in the next game to finish the match in two hours and 28 minutes.
Next up for Murray is No. 13 seed Lucas Pouille of France, who defeated Feliciano Lopez 6-7 (10), 6-3, 6-4.
To claim the No. 1 ranking, Murray needs to either win in Paris and have Djokovic lose before the final, or finish as the runner-up and have the Serbian lose before the semifinals.
The top-seeded Djokovic was in good form in his opening match, a 6-3, 6-4 win over Gilles Muller.
"It was great to start off the way I did today," Djokovic said. "Not an easy opponent because he serves and volleys, and I found the conditions quite quick. I think I managed to find the right rhythm, right pace from the very beginning... capitalized on the opportunities when they were presented. I was quite pleased with my serve. I think I got a lot of free points there. It was quite a good performance."
Next up for Djokovic is No. 14 seed Grigor Dimitrov.
Also Wednesday, American Jack Sock scored a 6-2, 6-4 upset of sixth-seeded Dominic Thiem of Austria. Sock fired eight aces and never faced a break point.
No. 11 seed Jo-Wilfried Tsonga had 14 aces in a 6-3, 6-4 win over Albert Ramos-Vinolas.
"I've won most of my tournaments here in France, so for me it's good to be here," said Tsonga. "It's really positive to be here in front of my home crowd."
Next up for Tsonga is No. 5 seed Kei Nishikori, who knocked off Victor Troicki 6-2, 7-5.
Gilles Simon delighted the local crowd by saving three match points to upset No. 10 seed Roberto Bautista Agut, 2-6, 6-1, 7-6 (6). The Spaniard led 5-3 in the final set and held three match points on his serve at 5-4. Simon saved all three, then won the tiebreaker.
"I'm really excited and satisfied with my form, especially these past few months," said Cilic.