The Black Sticks upset world No 4 England, a team ranked three places above them, in a penalty shootout after the score was tied 3-3 after both regulation and extra time.
A day after their female counterparts suffered heartbreak in their own penalty shootout in the gold-medal match against Australia, the men made no mistake.
They put away every one of their attempts, so that the one miss by England's Adam Dixon, who was thwarted by keeper Kyle Pontifex, proved decisive.
Burrows is one of three survivors from the 2002 campaign, along with vice-captain Dean Couzins and Hayden Shaw, and all three took part in the shootout.
Burrows admitted he couldn't look when the final penalty taker, Shea McAleese, stepped up.
When McAleese hit the back of the net, it was the signal for the start of New Zealand celebrations.
"It's right up there," Burrows said when comparing New Delhi bronze with Manchester silver.
"We've got a young group of guys coming through and a little bit of inexperience in these condition and with this crowd.
"It means a lot to take something away, especially after we've worked so hard over the past three or four weeks, slogging away in the heat. To go away with nothing would have been depressing."
The Black Sticks had arrived in India without two of their most seasoned players in midfield Ryan Archibald and Blair Hopping, who have both taken time out from international hockey.
Burrows said the team's consistency could have been better but he couldn't fault their fighting spirit, as shown when they rescued their last pool match against Canada 1-1 with a late Shaw goal.
"If we had lost that game, we would have been playing for whatever, fifth and sixth or something," he said.
"So we had to dig deep there and we had to dig deep today."
Couzins opened the scoring from the spot - the first of two penalty strokes New Zealand were given - after Blair Hilton was fouled by defender Glenn Kirkham.
Nick Haig made it 2-0 before England hit back with the first of Simon Mantell's three goals.
Another penalty stroke, after English keeper James Fair brought down Hugo Inglis, lead to Shaw converting for a 3-1 lead.
But when Steve Edwards was sinbinned after a collision with opposition skipper Barry Middleton, England took advantage of the extra man to level, with Mantell grabbing a quick brace to complete his hat-trick.
After neither side managed a golden goal, it was left to Pontifex, who had an excellent game overall with a string of saves, to perform the shootout heroics.
In the gold medal match, defending champions and world No 1 Australia overwhelmed hosts India 8-0.
