England led 20-8 at half-time but had to survive a second-half fightback which cut the gap to just two points before closing out a 33-21 semi-final win.
Second-rower Price and wing Purdy were both used as second-half replacements, while Wells has also gained tournament experience as England progressed to the final for a fourth time in six years at this level.
The win was England’s first against the Baby Blacks at age-grade level and now they meet Wales at the Stade de la Rabine in Sunday’s final.
Tries from flanker Matt Hankin, wing Anthony Watson and replacement prop Tom Smallbone helped England to the win, while fly-half Henry Slade added 18 points, and the defence stood up to an ambitious All Blacks attack.
Head coach Nick Walshe said: “The lads were absolutely unbelievable. From the first minute they put their bodies on the line.
“We knew New Zealand would want to play with the ball in hand, they made it very tough for us, but our defence was brilliant. We showed real character to stay in front.
“Wales will be a tough challenge. We obviously know them pretty well and they put in a massive performance against the South Africans, so credit to them and we look forward to seeing them on Sunday.”
Leicester Tigers scrum-half Scott Steele, meanwhile, was among the try-scorers as Scotland Under-20s beat USA 39-3 to earn a place in Sunday's play-off for ninth place against Samoa and also ensure their place in the World Juniors tournament again in 2014 in New Zealand.
England Under-20 (v New Zealand)
15 Jack Nowell (Exeter Chiefs)
14 Anthony Watson (Bath Rugby)
13 Harry Sloan (Harlequins)
12 Sam Hill (Exeter Chiefs)
11 Ben Howard (Worcester Warriors)
10 Henry Slade (Exeter Chiefs)
9 Alex Day (Northampton Saints)
1 Danny Hobbs-Awoyemi (Northampton Saints)
2 Luke Cowan-Dickie (Exeter Chiefs)
3 Scott Wilson (Newcastle Falcons)
4 Elliott Stooke (Gloucester Rugby)
5 Dominic Barrow (Leeds Carnegie)
6 Ross Moriarty (Gloucester Rugby)
7 Matt Hankin (Saracens)
8 Jack Clifford (Harlequins) (c)
Replacements:
16 Scott Spurling (Saracens) for Sisi 78 mins
17 Alec Hepburn (London Wasps) for Hobbs-Awoyemi 58 mins
18 Tom Smallbone (London Irish) for Wilson 68 mins
19 Tom Price (Leicester Tigers) for Stooke 66 mins
20 David Sisi (Bath Rugby) for Hankin 40 mins
21 Callum Braley (Bristol Rugby) for Day 66 mins
22 Ollie Devoto (Bath Rugby) for Howard 79 mins
23 Henry Purdy (Leicester Tigers) for Sloan 60 mins
