The win moves them up to fourth on the Super League standings, while for Waqa Blake and Jayden NIkorima's Bradford it is a third successive defeat, leaving them in 11th in their first year back in the top flight.
A Phoenix Steinwede try for the home side helped put them 10-2 ahead before Saints took control of the match.
Tries from Jacob Douglas and Sironen gave them the lead at the break before 28 unanswered points allowed them to stamp their authority.
Scores from Jack Welsby, with two, Joe Shorrocks and Hastings put St Helens firmly in the driving seat.
It was a first visit by Saints to Odsal since 2014 and marked a welcome return of the fixture after Bradford's period away.
In Toulouse, Josh Charnley scored four tries, with a first-half hat-trick, as Adrian Lam's Leigh saw off Toulouse 40-16 at a sweltering Stade Ernest Wallon to close up on the top six.
After seeing an early effort ruled out for the ball not being grounded under some last-ditch defence from Toulouse, the Leopards soon went ahead in the sixth minute when Charnley, the league's all-time record tryscorer, dived over in the right corner.
Charnley stretched to touch down another try in the 14th minute off a looping pass from Lachlan Lam out to the flank. Toulouse, looking to avoid a third successive Super League defeat of their own, tried to build some momentum ahead of the first hydration break, but were let down by a series of handling errors.
Leigh further extended their lead to 18-0 ahead of half-time, and pulled away from there.
Earlier on Saturday, Jai Field and Bevan French's Wigan Warriors beat Aidan Sezer's Hull FC 20-16.
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