Kashyap, who had reached the quarterfinals stage in London Olympics, won 24-22 20-22 22-20 in a marathon opening round match that lasted an hour and eight minutes.
Olympics bronze medallist Saina Nehwal also opened her campaign on a positive note, prevailing over Aprilla Yuswandari of Indonesia in straight games to reach the second round of women's singles event.
World number three Saina took time to get into the groove but went from strength to strength as the match progressed to notch up a 22-20 21-8 win over Aprilla at the Hong Kong Coliseum at Kowloon.
Ajay Jayaram also reached the second round after seeing off Zhengming Wang of China 22-20 23-21 in another men's singles match.
Kashyap showed tremendous resilience and a never-say-die attitude and matched Jorgensen with his repertoire of strokes during the thrilling match.
Trailing 0-3 initially, Kashyap turned the tables on Jorgensen and moved into the break at 11-8. The Dane, however, clawed back 13-13, but the Indian fought tooth and nail to nose ahead.
In the second game, Jorgensen and Kashyap traded points right from the start but it was the Dane who had enjoyed the lead most of the time and eventually succeeded to staving off the challenge from the Indian to bounce back into the contest.
The decider turned out to be a cliff-hanger as Kashyap showed nerves of steel and didn't let go off opportunities and despite being pushed to a 2-5 deficit early on, wrestled his way to lead 11-10 at the break.
After the breather, Kashyap slowly widened the gap and moved to 19-15. Jorgensen clawed back to 20-20 but it was the 26-year-old Indian who had the last laugh.
In other matches, Pranaav Chopra and Siki Reddy also reached the second round with a 21-19 21-15 victory over Phillip Chew and Jamie Subandhi of the United States in the mixed doubles competition.
Akshay Dewalkar and Pradnya Gadre lost to sixth seed Fran Kurniawan and Shendy Puspa Irawati 18-21 21-23 in mixed doubles, while men's doubles pair of Tarun Kona and Arun Vishnu bowed out with a 18-21 14-21 loss to Chris Adcock and Andrew Ellis.
Tarun and Ashwini Ponnappa lost their opening match against top seed Chinese pair of Chen Xu and Jin Ma 21-15 21-16 in mixed doubles.
Returing to action after skipping the China Open, Saina was equally good in her smashes as net play as the Indian registered as many as 15 smash winners and equal number of net winners in the 33-minute contest.
After gaining a narrow 3-1 lead early on, Saina lost the lead briefly when Aprilla went 8-6 ahead. However, the Indian restored her lead, moving to 14-10, before the Indonesian clawed back 20-20.
But Saina did not give any chance to Aprilla after that and closed the opening game in her favour with two consecutive points.
In the second game, the third seeded Indian was trailing 1-4 initially but came back in style, registering eight straight points to move to 9-4. She then reeled off another 10 points and was 19-6 ahead in a jiffy. She closed the issue in her favour comfortably after that.