A topsy turvy game saw Rimington progress through to the next round of the Craven Cup, but not without a massive scare as the visitors ran them close at the end.
Cononley took the lead inside two minutes when keeper Graeme Kershaw came to claim a cross, but the ball held up in the strong wind and the forward nipped into head over him into the empty net.
This shook the home side, and they struggled to regain composure for a while.
Matt Bagot was the one who showed the way with his darting runs, and he levelled matters with a cool finish.
Rimington now started to tick, with Charlie Collinge pressing forward from right back, and David Robinson showing pace down the flank, and from one of his runs they won a corner, the cross finding Martin Aspinwall at the back of the box, and he brilliantly lobbed the keeper to find the top corner.
Oli Devenney thundered in number three from the edge of the area, and then Collinge cut inside before curling home a beauty with his left foot on 40 minutes.
However, the visitors replied with a stunning goal of their own before half-time – the strike from 25 yards appeared to be sailing over, but the wind slowed it down enough for it to sneak under the bar for 4-2.
This seemed to energise them, and shortly after the break they pulled another back with the goal of the match, a midfielder from 30 yards out smashing the ball into the top corner, giving Kershaw no chance.
Rimington were stung into action again, Ben Allen and Bagot pulling the defenders all over, and the latter pounced on a mistake by the keeper to score his second for 5-3.
He could have grabbed his hat-trick when he hit the woodwork.
The visitors had their keeper to thank as he made three one-on-one saves to keep them in the game, but was powerless to prevent Devenney getting his second when he shrugged off a challenge before unleashing a shot into the net via a post.
With 10 minutes to go, the visitors were given a lifeline as Jonny Lincoln turned and slipped on the edge of his own area, and the striker stroked the ball into the corner for 6-4.
Shortly after, a cross into the box struck Jack Cook and flew into the net past the wrong-footed Kershaw and set up an exciting last five minutes.
Matt Garner should have sealed the game when through on goal but fired horribly wide.
Manager David Cook was a relieved man at the end: “To be fair, you have to credit the opposition who kept going until the end, and our defensive lapses didn’t help us, but I’m pleased with some of our attacking play, and we could have scored 15 on another day!
“We are in the hat and that’s what counts, but we have to tighten things up for sure.”
Saturday sees a change to the fixtures, with a home game against Whinney Hill.