He replaces his party colleague Arlene Foster, who held the finance job until she became first minister on Monday after Peter Robinson stepped down.
Mr Storey, who held the social development portfolio, will be replaced by Lord Morrow.
Mr Storey said he took the responsibility of managing the public finances "very seriously".
"I do not underestimate the challenge ahead with tight budgets and greater demands on our public services but I am determined to do my bit to improve services, raise productivity and reduce costs in the public sector," he said.
The finance minister said he would continue the work already under way to ensure that the rate and date set for corporation tax could be devolved from April 2018.
Mrs Foster had been expected to name a replacement rather than carry out a full ministerial reshuffle.
One reason given was that the Northern Ireland Assembly will break up in March to allow an election campaign to begin.
Downing Street released a statement on Tuesday to say that Prime Minister David Cameron had phoned Mrs Foster to congratulate her on her appointment.
"The prime minister reflected on the milestone of the first female first minister being appointed, as well as the ministerial experience and excellent working relationships which Mrs Foster will bring to the role," the statement said.
"Mrs Foster and the prime minister agreed on the importance of implementing the Fresh Start and Stormont House agreements and the executive being able to focus on the economic issues which affect the everyday lives of people in Northern Ireland.
"The first minister and the prime minister agreed to maintain a close working relationship and regular dialogue to ensure progress in building a more prosperous and shared future for the people of Northern Ireland."
The DUP's Mervyn Storey has been appointed as finance minister for Northern Ireland.