The Canaries kicked off attacking the River End goal and in the seventh minute Gary Doherty rattled the Saints crossbar with a fierce left-foot volley from Simon Lappin's free-kick on the City right.
David Marshall had no trouble with Jones' header and at the other end Kelvin Davis did well to cut out a low right-wing cross from Jon Otsemobor.
Former Canary Youssef Safri was booked in the 20th minute for a late tackle, before Jhon Viafara fired into the sidenetting.
Jones had a shot cleared off the line by Adam Drury for a corner. Safri fired in the flag kick, Marshall appeared to be impeded and the ball fell kindly for Jones, who found the inside of Marshall's right-hand post with a low right-foot shot from eight yards.
The goal lifted the Saints and they finished the half looking much stronger than the Canaries who at times lived dangerously in their own penalty area.
Five minutes into the second half Marshall did well to save from Grzegorz Rasiak.
The City keeper then saved Andrew Surman's shot preventing it from crossing the line at the second attempt.
This save paid off because a back pass from Chris Makin led to an indirect free-kick. This was laid off by substitute Mark Fotheringham to Cureton whose right-foot shot found the top right hand corner of Davis' goal.
The Saints keeper then pulled off two acrobatic saves to deny Cureton and Fotheringham.
However, there was nothing he could do to keep our Cureton's winning goal.
Lee Croft crossed from the right and Cureton controlled the ball on his chest before hitting a fierce right-foot volley from 12 yards. beating Davis as the keeper dived to his right.
It was the Saints then who looked the most vulnerable as Norwich tried to kill the game with a third goal, but they had done enough to secure a victory.