Kumar’s scoring was inferior to Veenstra’s, but his finishing proved crucial, boasting a 75% checkout success and hitting all his last seven doubles.
“I don’t know what to say right now. I’m overwhelmed, I’m happy,” said Kumar, who is nicknamed ‘The Royal Bengal’.
“If you dream it, anything is possible. I’ve dreamed of this ever since I saw Dennis Priestley win the World Championship (in 1994).”
Asked about what his victory could do for darts in India, he said: “I’ve opened the floodgates to a billion of them.
“I’m sorry, 10 years down the line if you have eight people in the World Championship walking on to Bollywood music, don’t blame me.”
Veenstra hit five 100-plus finishes, with his highest coming at 144. Kumar was not as flamboyant with his scoring, but proved to be incredibly consistent as the crowd roared him on.
Three-time BDO world champion Glen Durrant said on Sky Sports: “It was one of the greatest games I have ever commentated on.
“It’s not just the quality of the match, it’s the impact for Nitin Kumar and what it does for Indian darts.”
Two seeds exited the competition on Sunday – world number 23 Dimitri van den Bergh and 27th-ranked Ritchie Edhouse.
Belgium’s Van den Bergh was out of sorts from the first dart and was comfortably beaten 3-0 by Scottish debutant Darren Beveridge.
In the afternoon session, Edhouse was knocked out by Jonny Tata, who was also making his first appearance.
Former European champion Edhouse was stunned 3-0 by the New Zealander, to taste defeat at this stage for the fourth time in a row.
But the same fate did not follow fellow seed Joe Cullen after he eased past Bradley Brooks with a 3-0 win.
The world number 32 was in fine form to produce his second-highest average on the world stage with 99.33.
Another debutant, Englishman Dom Taylor, progressed to the second round with a 3-0 win over Sweden’s Oskar Lukasiak.
Meanwhile, the evening session saw the Netherlands’ Lukas Wenig beat Wesley Plaiser 3-1 while England’s James Hurrell beat Stowe Buntz by the same scoreline.