The 25-year-old Lahiri birdied the 18th hole from 10 feet in regulation play to force extra-time and then claimed his third Asian Tour win with a four-foot birdie conversion on the same hole at the Delhi Golf Club.
Khan, 22, gave himself a great chance of a maiden title on the region's premier Tour with a superb final round of six-under-par 66, which included a closing birdie, but he could not prevent a dominant Lahiri, who signed off with a 68.
Shiv Kapur of India settled for fourth place after a 70, while Thailand's Pawin Ingkhapradit and Australian rookie Matthew Stieger ended in tied fifth place.
"It's not sunk in. I'm still in disbelief that I made the putt in regulation and got into the play-off. After that, in the next 15 minutes, I didn't know what happened," he said.
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"Unfortunately for Rashid, it didn't work out for him. I feel bad for him as he doesn't have a card on the Asian Tour. He's got a bright future ahead of him," added Lahiri, who ended the week on 15-under-par 273 with Khan.
Khan, playing in the penultimate group, brilliantly birdied his 72nd hole from three feet but could do very little as Lahiri coolly matched his closing birdie from 10 feet to force the play-off.