The 20-year-old launched her campaign in the women's S14 100m butterfly, a class for swimmers with intellectual impairment.
Chan was third at the turn of the pool, before picking up speed to come back second with an Asian record time of 1:03.70.
She was 0.7 seconds adrift of Britain's Poppy Maskill, who stormed to gold while setting a world record time on her Games debut.
Valeriia Shabalina of Russia, competing under a neutral banner, rounded up the podium.
"There's still some distance from the gold medal, but I think I have done very well already," said an emotional Chan after her race.
"I told myself to come and enjoy the race, and give it my all. Out of many races, I enjoyed this the most."
In a statement, Chief Executive John Lee congratulated the SAR swimmer on claiming a silver medal in Paris, saying she showed great perseverance.
Winning starts for other SAR athletes
To the courts, and Tokyo silver medallist Chu Man-kai won opening singles group match in badminton SH6 – an event for shuttlers with short stature.
He beat Tokyo bronze winner Krysten Coombs of Great Britain 21-13, 19-21 and 21-15.
Chu will next play India's Sivarajan Solaimalai on Friday.
A number of bowlers were also victorious, including six-time Paralympian Leung Yuk-wing who defeated his Brazilian counterpart 8-1 in the men's individual BC4 preliminary round.
World number two Ho Yuen-kei beat her South African opponent in the women's individual BC3.
China tops medal table
Team China topped the medal table at the end of day one with four gold medals and one silver, with wins in swimming and cycling.
Track cyclist Li Zhangyu claimed the nation’s first gold, edging out compatriot Liang Weicong in the men's C1 3,000m individual pursuit.
Swimmer Chen Yi set a world record in the women’s 50m freestyle S10, while 19-year-old Jiang Yuyan made it to the top of the podium in the women's 50m S6 freestyle.
Dutch domination in velodrome
The first gold medal at this year's Paralympic Games went to Caroline Groot of the Netherlands.
The seven-time world champion rode to victory in the women's cycling track C4-5 500m time trial, having earlier refreshed the world and Paralympic records in the same race.
