The Ashes heroes are replaced by Stuart Broad and James Anderson in a new-look pace attack.
Harmison took 1-121 in the first Test defeat in Hamilton, while Hoggard (1-122) was equally disappointing.
"It was a tough call. We felt the attack needed a bit of a shake-up,"
said captain Michael Vaughan.
"We had a gut feeling that we just needed to make a change and we've gone with two of them.
"It's a great opportunity for Jimmy and Broady to come in and try and establish themselves at this level.
"I also hope Matthew Hoggard and Steve Harmison really fight and try to get their places back because that's going to create a healthy environment as well."
Harmison, 29, who retired from one-day internationals 18 months ago, has struggled for form since England's 2005 Ashes success.
But Hoggard, 31, has been one of England's most potent attackers and his omission is more surprising.
This is the first time the duo have been dropped since helping England regain the Ashes nearly three years ago and leaves the tourists with an inexperienced attack with only 32 caps between Ryan Sidebottom, Anderson and Broad.
But Vaughan said:
"It's quite an exciting time. I'm leading a new attack into a Test match.
"It certainly looks very exciting, even if it is a little bit lacking in experience but I think they have enough talent to take them through and put the New Zealand batsmen under a lot of pressure."
Nottinghamshire's Broad claimed six wickets in the first three ODIs against New Zealand and his ability with the bat - he started his cricketing career as an opening batsman - will be an added bonus.
Asked whether England had considered making a change in the batting line-up, Vaughan added:
"Not really...a change was talked about but we thought they deserved another opportunity to try and put it right this week."
England: Cook, Vaughan (capt), Strauss, Pietersen, Bell, Collingwood, Ambrose (wkt), Broad, Sidebottom, Panesar, Anderson.
New Zealand: Bell, How, Fleming, Taylor, Sinclair, Oram, McCullum (wkt), Vettori (capt), Mills, Gillespie, Martin.
Umpires: RE Koertzen (SA) & SJ Davis (Aus).
Third Umpire: EA Watkin (NZ).
Match Referee: Javagal Srinath (Ind).