The agreement with the ultra-conservative Northern Irish party was signed in May's Downing Street office after more than two weeks of negotiations following her disastrous showing in the June 8 general election.
Under the terms of the deal, Northern Ireland will receive an extra £1.0 billion (1.1 billion euros) from the state over two years in exchange for DUP supporting May's Conservatives.
The Conservatives have 317 seats in the 650-seat parliament after the election and need the support of the DUP's 10 MPs to be able to govern.
The deal with the DUP will prove controversial because of the party's opposition to gay marriage and abortion and concern that an agreement could upset the fragile balance of the peace process in Northern Ireland.
The deal is a "confidence and supply agreement", meaning that the DUP will only guarantee to support the Conservatives in confidence and budget votes.
For any other measures support would be on a vote-by-vote basis, the text of the agreement said.
"I welcome this agreement which will enable us to work together in the interests of the whole United Kingdom," May said in a statement.
'Shoddy little deal'
Foster said: "This agreement will operate to deliver a stable government in the United Kingdom's national interest at this vital time."
The DUP supported Brexit but has emphasised the need to keep the border with the Irish republic open, and Foster said the deal would back a Brexit process "that supports all parts of the United Kingdom".
Foster said the extra money would be spent on infrastructure, health and education, benefitting the whole of Northern Ireland after concerns voiced by the republican Sinn Fein party, the DUP's rivals.
The deal will face its first test in parliament with a confidence vote expected on Thursday.
The main opposition Labour Party has said it will push for another general election.
Liberal Democrat leader Tim Farron immediately slammed the agreement as a "shoddy little deal".
"The nasty party is back, propped up by the DUP," he said in a statement.
Discussions between the Conservatives and the DUP began immediately after the election, stirring up further resentment against embattled May.
Some DUP representatives have been criticised in the past for homophobic comments, climate change denial statements and sectarian rhetoric.
Ireland's former premier Enda Kenny has warned that a deal with the Protestant and pro-British DUP could upset Northern Ireland's fragile peace.
London's neutrality is key to the delicate balance of power in Northern Ireland, which was once plagued by violence over Britain's control of the province.
Northern Ireland's political parties are still locked in negotiations to form a semi-autonomous power-sharing executive for the province, nearly four months after local elections there.
If the parties cannot come to an agreement by Thursday, Northern Ireland may be returned to direct rule from London.
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