He narrowly beat his rival Sir Jeffrey Donaldson, by 19 votes to 17.
In his acceptance speech, Mr Poots vowed the DUP would be the “authentic voice” of Unionism under his leadership.
But he said he would reach out to other Unionist parties to oppose the controversial NI Protocol, which, under Boris Johnson’s Brexit deal, has effectively created a border down the Irish Sea.
He urged Unionists to band together to fight this “massive challenge” and for an end to the “Unionist bickering” like there had been in the past.
Northern Ireland’s second largest Unionist party, the Ulster Unionists, have announced their own leadership contest in recent days in what is seen as a bid to capitalise on turmoil within the DUP.
Ms Foster was ousted after an internal power struggle by party colleagues unhappy with her leadership.
She will step down as leader on May 28 and as Northern Ireland’s first minister at the end of June.
The DUP has faced growing anger from sections of the Unionist community over its handling of Brexit and the NI Protocol in recent months.
The party has been accused of squandering its influence at Westminster, particularly during its confidence and supply deal with the Conservative government before the 2019 general election.
There was also controversy over Ms Foster’s decision to abstain in a vote on gay conversion therapy.
Speaking to The Times in 2007, Mr Poots, whose father was one of the founding members of the DUP, said: “My view on the earth is that it’s a young Earth. My view is 4000BC.”
Seen as from the more religious wing of his party, Mr Poots also has a reputation for developing good working relationships with other politicians.
His election comes just months after he underwent a major operation for kidney cancer, which was discovered by chance when he had surgery on a burst appendix before Christmas.
Only the DUP’s eight MPs and 28 Stormont Assembly members were eligible to vote in the leadership election.
Mr Poots will be leader designate until Ms Foster formally stands down.
Paula Bradley was elected the party’s new deputy leader. She defeated Gregory Campbell by 18 votes to 16.