Lee Smith and Danny Kirmond scored two tries apiece for the hosts, but Saints responded with a Francis Meli hat-trick and Paul Wellens double.
That saw them claw back a deficit which had at one point reached 14 points, but it was all in vain as Sykes had the final say.
The Wildcats had tried desperately to land the decisive blow, but both Smith and Sykes missed with gilt-edged drop-goal opportunities before the latter nailed the winning point right on the hooter to secure a famous victory.
Andy Raleigh and Kyle Wood also scored tries for the hosts while Puletua crossed for St Helens.
Hull FC dismantled a powerful Catalan Dragons side at the KC Stadium in a convincing 30-10 win that displayed their credentials of claiming a top-four spot.
The Black and Whites closed the gap between themselves and the French side to two points as Ben Crooks grabbed his fourth try in as many games and Jamie Foster scored 14 points.
Willie Manu grabbed the first try of the match before Vincent Duport hit back for the Dragons to level it up. Either side of the break Tom Lineham, Tom Briscoe and Chris Green extended the home side's lead to 24-6 and, after Duport pulled a try back, Foster sealed it with a late try in the corner.
Luke Dorn claimed an impressive four-try haul to help London Broncos to a long-awaited 42-20 away win over fellow Super League strugglers Castleford.
The utility player went over for a first-half hat-trick and added a fourth after the break as the Broncos were triumphant on the road in Super League for the first time since last March.
This victory lifted them level on points with Widnes but above them due to their superior points difference, while Castleford's woes continue after a fourth straight defeat.
- Danny Kirmond
- Francis Meli
- Paul Wellens