If England wins the series by a two-test margin, it will overtake India as the top-ranked team in the world.
Anderson took 5-65 as India, which began the fifth day on 80-1 in pursuit of an unlikely 458 to win, collapsed in the evening session at Lord's, with Broad chipping in with 3-57.
"That was as close to a perfect bowling performance as we've had in the last two years," England captain Andrew Strauss said. "We know how hard you have to work to get your wickets here. Today, when they were five down at tea, we had to work really hard to get one end open and when we did that we finished it clinically."
India captain Mahendra Dhoni felt his side had struggled to compete after swing bowler Zaheer Khan was struck down by a hamstring injury that over-burdened the bowling attack.
"I think if we'd had an extra bowler things could have been different," Dhoni said. "It became quite tough for us to manage the bowlers after losing Zaheer Khan on the first day. If we had had Yuvraj Singh or Virender Sehwag, it would have given us more options."
India are sweating on Khan's fitness for the second test, which begins at Trent Bridge on Friday.
"It's a long series, so we don't really want to risk any individual who's close to 80 or 90 percent fitness," Dhoni said.
Suresh Raina led India's rearguard action, making 78 from 136 balls with 10 fours. VVS Laxman scored 56, while Sachin Tendulkar was out for 12 in possibly his final match at the home of cricket.
Tendulkar fell an hour into the afternoon session, two balls after being reprieved when a delivery from Anderson hit the toe end of his bat and he was dropped by Strauss at first slip.
However, to Strauss's relief, Anderson removed Tendulkar by trapping him plum lbw, ending an innings in which he had faced 38 balls and went 47 minutes without scoring.
Before another run had been added, Raina was given not out by the third umpire in a reviewed stumping appeal by Matt Prior off Graeme Swann. Replays showed Raina's toe may just have been in his crease and he was given the benefit of the doubt.
Raina looked like he might help rescue India as he forged a 60-run partnership with Mahendra Singh Dhoni, until the Indian captain was caught behind off Chris Tremlett for 16 early in the evening session.
Eoin Morgan dropped a simple chance to catch Harbhajan Singh on 11 off Broad, whose next delivery thundered into the batsman's pads but he was ruled not out by Billy Bowden.
Replays showed this to be the umpiring crew's first serious mistake of the test, but Harbhajan's reckless batting rendered it academic. In the next over, he swished a ball from Anderson high into the air and was caught by Tremlett at mid on for 12.
Praveen Kumar made just 2 before he was clean bowled by Broad, and though Raina then survived a referral for caught behind off Anderson, he was out at the end of the same over, caught by Prior.
It was left to Broad to finish India off five balls later, when he had Ishant Sharma lbw to spark raucous celebrations.
On a day of unrelenting tension, fans had started queuing at 2am for the 20,000 tickets available, yet the ground was still only around half full when play began nine hours later, with thousands still locked outside.
Some stands were opened only after India resumed on 80-1, although they were filling rapidly by the time the first chance of the day went begging.
A ball from Tremlett looped up off Rahul Dravid's bat and pad, just beyond the reach of Ian Bell at short leg.
It was a short-lived reprieve for Dravid, who wafted at an outswinger from Anderson and was caught behind for 36.
Gautam Gambhir, who was batting with a severely bruised elbow, edged Broad's first ball but it fell short of the slip cordon and India received four overthrows when Strauss's shy at the stumps flew to the boundary.
England then wasted a review when the third umpire upheld Bowden's decision not to give VVS Laxman out caught behind to Broad.
