Speed never gave up hope of a resolution
The announcement has put an end to the saga that had threatened to cause the cancellation of the entire three-Test series.
The International Cricket Council (ICC) would have declared the match unofficial if the banned Sehwag took to the field.
But in return for India not selecting Sehwag, the ICC has agreed to review the way match referee Mike Denness imposed penalties during this month's second Test against South Africa.
It was in that match that Denness disciplined six Indian players but no South Africans.
The ICC has also agreed a referees' commission to clear up any uncertainty relating to their decisions.
At Lord's on Friday, ICC chief executive Malcolm Speed was visibly relieved that the saga had been brought to an end.
"I am very pleased," he said. "Let's hope we have a great Test series between these two great powerful cricketing nations and let's hope that the great game of cricket comes out stronger.
"Whenever there's an issue like this there are strong emotions. But I'm happy with the result and I think Mr Dalmiya is happy with the result."
England's cricket board (ECB) said it was delighted an agreement had been reached.
"I am delighted to hear this news. The decision is a victory for common sense," said chairman Lord MacLaurin.
"It will strengthen the ICC's authority and integrity as the world governing body for cricket, but the family of cricket is the real winner."
Tim Lamb, ECB chief executive, said, "This is good news for the millions of cricket followers worldwide and England's supporters.
"There is huge expectation about the series and I am delighted that the tour can now proceed as planned, and that the England team can continue their preparations for the first Test in Mohali without any further distractions."
England would have had no option but to withdraw from the first Test if 23-year-old batsman Sehwag had been selected.
Sehwag has played in just two Tests but hit a century on his debut against South Africa.
Then, in the second match at Port Elizabeth, he was one of six Indians disciplined by match referee Denness.
Denness, who has invoked the wrath of millions of Indians, banned Sehwag for one match.
However India insisted Sehwag served that ban in the match at Centurion Park, which was declared an unofficial Test after both teams agreed to drop Denness as referee.
And even now that the flames of fury have been dampened down, Indians are still angry about Denness.
Niranjan Shah, secretary of the Indian cricket board, told BBC Five Live: "We have come to a good solution - that the cricket must go on.
"The ICC has also come to a good solution - to establish a report on the match referee, because he has given biased decisions and been inconsistent."