The IPO is the first this year by a so-called technology "unicorn" - start-ups valued at more than $1bn (Â£700m).
Twilio's platform that enables developers to incorporate phone calls and text messages into their apps.
Shares were priced at $15 but closed at $28.19, valuing the company at more than $2bn.
Its software is used by companies including OpenTable, WhatsApp and Uber, which uses the technology to withhold the actual mobile numbers of drivers when they call passengers.
The appetite for Twilio shares has revived investors' hopes in the US technology sector.
Analysts believe that its successful debut market could be a positive sign for other "unicorns" looking to go public.
Jack Ablin, chief investment officer at BMO Private Bank, said: "[This year] has really been an arid desert in new issuance and equity ... I'm encouraged that the success here will spawn other IPOs [initial public offerings]."
Shares in software firm Twilio have soared almost 88% on their first day of trading in New York.