After the two clubs agreed a fee believed to be around £1.5m the 27-year-old travelled to Glasgow on Thursday night to discuss personal terms.
Murphy has penned a a three year deal and is expected to join his new team-mates on their tour of North America, with Celtic facing Manchester United and Seattle Sounders over the weekend.
Murphy becomes Neil Lennon's fourth summer signing, following the captures of left-back Charlie Mulgrew, right-back Cha Du-Ri and midfielder Joe Ledley.
Hoops boss Lennon revealed he had been an admirer of Murphy for some time.
He's quick, he's six foot two or three and I've watched him for a few years, Lennon said. I think he could take the hits for us up here.
The striker joined Sunderland from League of Ireland side Waterford in 2005 after being named the PFAI Player of the Year, initially spending a brief loan period with Sheffield Wednesday.
Murphy was also loaned to Championship side Ipswich Town last season where he scored six goals in 18 appearances under former Celtic midfielder and ex-Sunderland manager Roy Keane.