Peters and his fellow opener Robert Key put on 100 for the first wicket to give England the perfect platform for victory, which was achieved with four overs to spare after being set a target of 242.
Peters played cautiously at first before punishing the loose ball, of which there were many, and he was finally dismissed for 107 off 125 balls with 12 fours.
Captain Owais Shah played his best cricket of the tournament in making an unbeaten 54 as the winning run came from a leg bye, ensuring England became the second winners of the tournament, Australia being successful on their home soil in 1988.
"This is probably the best innings I've ever played in my short career," said the 19-year-old Essex opener, who was the youngest Englishman to make a hundred on his first-class debut, in 1996.
"To chase 242 was tricky and we would have preferred 200 to 210 but Robert Key and myself set about it in the perfect way. It was just up to one of us to bat through and it turned out to be me."
A partnership of 99 for the sixth wicket helped dig New Zealand out of trouble to finish on 241 for 6. They were helped by 22 wides and five no-balls in a generally ill-disciplined bowling display by England.
Wicketkeeper Peter McGlashan smashed 53 off 38 balls but was dropped by Giles Haywood and Key, while James Franklin hit an unbeaten 56.
Having been put in to bat, New Zealand got off to a flier, adding 71 for the first wicket in less than 13 overs, assisted by some wayward bowling but Haywood and Graham Napier stopped them in their tracks and took five wickets between them.
Haywood bowled a tight line and the batsmen struggled against his nagging medium pace to finish with 3 for 18 off 10 overs.
England Under-19 won toss
NEW ZEALAND UNDER-19
J Marshall c Wilton b Napier 26
D Kelly c Wilton b Napier 26
*J Englefield c Schofield b Haywood 1
H Marshall c Schofield b Haywood 14
L Vincent b Haywood 19
J Franklin not out 56
P McGlashan b Franks 53
R West not out 13
Extras (lb6 w22 nb5) 33
Total (for 6, 50 overs) 241
Fall: 1-71 2-77 3-77 4-107 5-118 6-217.
Did not bat: K Mills, B Martin, T Anderson.
Bowling: Franks 7-0-49-1; Logan 7-0-44-0; Napier 9-3-39-2; Swann 6-0- 29-0; Haywood 10-5-18-3; Powell 8-1-34-0; Schofield 3-0-22-0.
S D Peters c West b Anderson 107
R W T Key c Vincent b Franklin 27
P J Franks c Mills b Martin 16
*O A Shah not out 54
G P Swann not out 22
Extras (b1 lb6 w7 nb2) 16
Total (for 3, 46 overs) 242
Fall: 1-100 2-122 3-211.
Did not bat: C P Schofield, G R Haywood, G R Napier, J C Powell, N J Wilton, R J Logan.
Bowling: West 8-0-33-0; Mills 10-0-54-0; Martin 10-0-38-1; Franklin 5- 0-35-1; Anderson 10-0-59-1; J Marshall 3-0-16-0.
Umpires: S Deenik and J Cloete.
England Under-19 won by seven wickets.Reuse content