Overnight leader Rickie Fowler missed out on an automatic Ryder Cup place after carding a three-over round of 74 on Sunday to finish joint seventh.
Fowler needed to finish third and will now hope to be a wildcard pick on Davis Love's team.
Reed, on the losing team at Gleneagles in 2014, said: "I want sweet revenge, just like our whole team does."
Europe have won the competition three times in a row, and eight times in the last 10.
Reed, 26, beat fellow American Sean O'Hair and Argentina's Emiliano Grillo by a shot in New York to go top of the PGA points standings.
Scotland's Martin Laird ended two under par, while England's Olympic champion Justin Rose and Northern Ireland's Rory McIlroy finished on level par.
Reed, who represented the USA at the Presidents Cup in South Korea and at the Olympic Games in Brazil, added: "I finally get to play for the United States in the United States and I cannot wait to see how the fans are, and just have the whole crowd on our side this time."
Fowler's late slip-up saw 2015 Open champion Zach Johnson claim the final automatic spot.
1. Dustin Johnson
2. Jordan Spieth
3. Phil Mickelson
4. Patrick Reed
5. Jimmy Walker
6. Brooks Koepka
7. Brandt Snedeker
8. Zach Johnson
Captain Davis Love has four wildcard picks and will choose three after the BMW Championship on 11 September.
Love will make his final selection two weeks later after the Tour Championship, only five days before the Ryder Cup starts on 30 September.
Among those in contention for a spot, along with Fowler, are Bubba Watson, who finished ninth in the standings, and JB Holmes.
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The ticket for the Grosvenor in Norwich was bought for seven shillings and sixpence and signed at the gig by John Lennon and Ringo Starr.
It also states the band were supported at the now defunct venue on Prince of Wales Road by local band Ricky Lee and the Hucklebucks.
The ticket is being auctioned by Bonhams on 3 July.
At the time of the gig The Beatles were celebrating their first number one single From Me To You, which stayed at the top of the British singles chart for seven weeks.
Bonhams said if the ticket also contained the signatures of Paul McCartney and George Harrison, it would have been worth nearer £4,000.

That the UK is still 'bold' but 'dependable', 'outward looking' and a bastion of free trade. She has done that in private sessions with several leaders and a press conference with President Obama.
But very frank words from the United States, Japan and Donald Tusk, the President of the European Council, will steal plenty of the headlines instead.
Remember she was an advocate, albeit a pretty quiet one, of staying in the EU in the first place, with the economy one of her main concerns. So Downing Street says she hasn't been surprised by the warnings about the future.
And perhaps dealing with public expressions of sentiments that have been already expressed in private isn't really the problem.
Perhaps the real difficulty for the now not so new prime minister is that the biggest demands on her abroad are the same as those at home. Other countries want more details of her plans for Britain's life after the EU.
But with no consensus, or anything like the details of post-Brexit Britain from government, there isn't that much that she can really tell them. The political danger of such a relatively blank page is that others will fill the space.
Patrick Reed won his first tournament of the season at The Barclays to seal his spot on the USA Ryder Cup team.
[NEXT_CONCEPT]
A signed Beatles ticket from a concert in Norfolk in 1963 is expected to fetch up to £2,000 at auction in London.
[NEXT_CONCEPT]
Theresa May's intention at this summit was to reassure other countries that life after the EU might be tricky to work out, but in the end, all will be just fine.