Fast-medium bowler Broad took a Test-best seven wickets for 44 runs as New Zealand were dismissed for just 68 in pursuit of a target of 239 to give England victory by 170 runs, with more than a day to spare.
Anderson, set to be England's key bowler when they bid for a third straight Ashes series win over Australia later this season, took five wickets in New Zealand's first innings 207 -- made in reply to the hosts' modest 232.
In the course of that haul, Anderson became only the fourth England bowler to take 300 Test wickets after Fred Trueman, Bob Willis and Ian Botham.
Concerningly, in the midst of a resounding success, England only managed 213 in their second innings with Broad's 26 not out Sunday a reminder of his batting ability.
"It's a pretty special day for us, we wanted to make more of a contribution with the bat but we knew the ball was moving around," Broad, who exploited the conditions superbly, said at the post-match presentation.
"I've come away with seven wickets but Jimmy Anderson was the leader of this attack, he got five wickets in the first innings and I think he should be coming up to get this man-of-the-match award," the 26-year-old added in praise of his new-ball partner.
"It's a great way to start the summer, I love bowling here at Lord's," said Broad after topping his previous Test-best of seven for 72 against the West Indies at Lord's last year.
Meanwhile England captain Cook struggled to explain Sunday's dramatic turn of events.
"Cricket's a strange game and sometimes results like that happen. I think the extra heat today (Sunday) helped the ball swing, it was a good wicket but it kept doing enough, and Broady and Jimmy were outstanding," Cook told BBC Radio's Test Match Special.
For New Zealand this latest collapse, which saw them bowled out in under two hours Sunday and followed their 45 all out against South Africa at Cape Town in January, was particularly hard to take as they'd been in the match right until the closing stages.
"It's a tough one to explain, I thought we had parity or dominated for long periods but we took a step backwards today," said New Zealand captain Brendon McCullum.
"The margin flatters England, but they bowled superbly in that hour before lunch which ripped the heart out of us."
England themselves had wilted in the face of fine seam bowling by New Zealand's Tim Southee, who took six for 50 in their second innings for a match haul of 10 for 108.
"Tim Southee deserved more than we gave him," said McCullum.
New Zealand travel north to Headingley, where the second and final Test of this series starts on Friday, with injury doubts over both wicket-keeper BJ Watling (left knee) and spinner Bruce Martin (right calf).
"BJ Watling is a bit stiff and sore, we'll evaluate him over the next few days," McCullum, who deputised behind the stumps after Watling went off injured on Saturday, said.
"Bruce Martin is touch and go for the next game so we'll have to see."
Stuart Broad salutes 'leader' James Anderson as England hammer New Zealand
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