Trailing 12-2 at the break, Saints took full advantage of Joel Moon's dismissal to turn the game on its head with the help of tries from Louie McCarthy-Scarsbrook and James Roby.
Moon was given his marching orders for a shoulder charge on Mark Percival with an hour left and Saints saw off stubborn resistance from their Yorkshire opponents to eventually run out winners.
That put them on the brink of securing top spot, but Nathan Brown's men will have to wait to officially ensure the League Leader's Shield after Castleford Tigers returned to winning ways against Bradford Bulls on Sunday to move four points off the top with two games remaining.
Castleford, who lost to Warrington Wolves last time out in the Super League before tasting defeat to Leeds in the Challenge Cup final last weekend, ran in five tries to see off the Bulls 32-18.
It keeps alive Castleford's slim hopes of finishing the regular season top on what was a largely positive weekend of results for Saints, who also saw Wigan Warriors suffer a surprise 24-10 defeat to play-off hopefuls Widnes Vikings.
Widnes are close to ensuring a last-eight berth after picking up a shock win in what could prove skipper Jon Clarke's final home game before retirement.
Patrick Ah Van went over twice while man-of-the-match Danny Tickle kicked six goals to put Widnes close to extending their season.
High-fliers Warrington Wolves and Huddersfield Giants drew 24-24 at the Halliwell Jones Stadium to bolster Saints' title hopes, with the hosts roaring back from a 19-6 deficit at the break.
Giants captain Danny Brough kicked a dramatic last-gasp drop goal to ensure a draw, while Catalan Dragons hammered London Broncos 46-4.
Morgan Escare grabbed a hat-trick in the rout at the Stade Gilbert Brutus to keep the French side on course for a play-off spot.
Hull FC took the bragging rights against city rivals Hull KR by winning 28-0, running in five tries to leave KR's play-off hopes dangling by a thread after a defeat that coach Chris Chester described as "embarrassing, pathetic and soft".
Wakefield Wildcats are also in with a slight shout of the last eight after running in eight tries to comprehensively beat Salford Red Devils 42-6 in the final action of the weekend.