Former Northern Ireland international Feeney, 34, was previously the Exiles' assistant manager and replaces former boss John Sheridan, who left to join Oldham Athletic on Wednesday.
He will be assisted by former Bolton and Blackburn defender Andy Todd, who he previously worked with at Linfield.
Newport are 20th in the table, three points above the relegation zone.
Former Linfield manager Feeney spent a large spell of his playing career at Bournemouth, before later playing for Cardiff and Swansea.
He now becomes County's fifth manager in less than 12 months.
"It was an easy choice for me," Feeney told BBC Radio Wales. "It's a dream that it has happened.
"I want to hit the ground running and get results. This is a great area and a decent club."
Feeney left his role as manager of Belfast club Linfield in October to become Newport's assistant boss.
Meanwhile, Michael Flynn will continue is his role as first-team coach at the south Wales club.
Todd was assistant to Feeney's successor at Linfield, Northern Ireland's record goal scorer David Healy.
"We're really happy. It was an easy decision for us as a board of directors," Newport County director Gavin Foxall said.
"It gives us the stability the club needs at the moment."
Meanwhile Newport have signed Chelsea goalkeeper Mitchell Beeney on a one-month youth loan.
The 20-year-old is the son of former Brighton and Leeds United goalkeeper Mark Beeney.
Warren Feeney has been appointed as the new manager of League Two side Newport County.