The free-swinging Monfils broke Draper's serve in the fourth set as the clock moved past 11pm local time and saved five break points in an epic seventh game to move 5-2 ahead.
Former champion Naomi Osaka was sent packing from Indian Wells after losing in straight sets to Colombia's Camila Osorio.
Athletes from Russia and Belarus, Moscow's closest ally in the 2022 invasion of Ukraine, are barred from the Paris Games except for the few who will compete as neutral athletes without flags or anthems.
Serena Williams remains on course for a record sixth win at the Sony Ericsson Open after beating sister Venus 6-4, 3-6, 6-3 to set up a final meeting with 19-year-old Belarusian Victoria Azarenka. Azarenka defeated Russian Svetlana Kuznetsova 6-3, 2-6, 7-5 in Thursday's other semi-final to book her place in a top tier final for the first time.
Geopolitical tensions simmered at the US Open on Thursday as Ukrainian Marta Kostyuk refused the customary handshake at the net when Belarusian Victoria Azarenka defeated her 6-2, 6-3 in the second round.
Grand Slam champions Daniil Medvedev from Russia and Belarusian Victoria Azarenka will not be able to participate in the June 27-July 10 tournament.
'I played a good match, maybe he was a little bit tired from New York (where) he played an amazing tournament.'
'If grunting would win titles then everybody in the world would be grunting as loud as they can be.'
Seven-time Italian Open winner Rafael Nadal lost 7-6(7), 6-2 to Stanislas Wawrinka.
World number two Rafael Nadal dropped a set at the Barcelona Open for the first time in four years before advancing to the last 16, beating Argentine Leonardo Mayer 6-7(7), 6-4, 6-2 on Wednesday.
Photos from the matches played at the Miami Open on Monday
Second seed Sloane Stephens was shocked 2-6, 6-4, 6-2 by Germany's Andrea Petkovic in the second round of the Citi Open on Wednesday, disrupting the American's buildup to her US Open title defence.
Stanislas Wawrinka had to dig deep to beat Borna Coric at the Western & Southern Open in Cincinnati on Wednesday and conceded his thoughts had been far from tennis over the past few days.