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 Alexandra Palace 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.”
Earlier world number 27 Ritchie Edhouse was knocked out by debutant Jonny Tata.
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, with the Englishman hitting a brilliant 114 checkout.
