Tom Holroyd had a try disallowed a minute after half-time, unable to ground the ball before knocking on under pressure from Saints full-back Sailor.
Leeds did get their fourth try soon after though when Jack Bird’s beautiful backhanded offload freed up Ryan Hall to dot down his 357th career try.
Saints were on the board when Jack Welsby’s line break saw him race clear of Leeds’ midfield and full-back Lachie Miller for the match’s second excellent individual try.
But that proved to be just brief respite as Handley got on the scoresheet, latching on to a grubber kick from Connor, and another grubber from Connor found his half-back partner Croft, via the upright, for his 28th Super League try of the season.
Connor missed a penalty late on, striking the post, but his kicking game continues to impress with Leeds looking strong going into the upcoming play-offs.
Saints, meanwhile, will have to beat Leigh and Hull KR in their final two matches if they are to avoid missing out on the Super League play-offs for the first time.
Leeds Rhinos coach Brad Arthur told BBC Radio Leeds:
“I thought both our starts to the first half and the second half were pretty good. We did it with a good control.
“We had to stay composed and that was a good test for us. We had players all over the shop the last 15 minutes but they still had a pride in their defence which has been a bit questionable at times this season.
“If we can withstand the pressure for three, four sets, which we’re going to have to do against Wigan next week, it can take a bit of confidence out of your opposition.
“We’re up against a really good team who have been at the top of their game for a long, long time. So we’re excited, we’ve done the work, now we’ve got to be confident.”
St Helens: Sailor; Douglas, MacDonald, Welsby, Cross; Robertson, Lomax; Suluka-Fifita, Shorrocks, Klemmer, Davies, Wright, Host.
Interchanges: Walmsley, Humphreys, Keeley, Evans.
Sin-bin: Suluka-Fifita (15 mins)
Leeds: Miller; Hall, Bird, Handley, Newman; Croft, Connor; Oledzki, O’Connor, Jenkins, Watkins, Hankinson, Cassell.
Interchanges: Holroyd, Smith, Levi, Lodge.
Referee: Marcus Griffiths.