On a sunny evening at Saughton Enclosure, the two teams served up an entertaining encounter, which had everything but goals up until Connachan's 65th minute strike.
The first opportunity of the game fell to Hibs' Jonathon Grotlin, who found space on the edge of the box and fired a strike towards goal, but Sean Brennan in the Tynie goal was equal to the task and collected easily.
Tynecastle had the best chance of the half to equalise when Greg Binnie's teasing ball into the area found Kyle Mitchell, who cannoned a hard shot off of the Hibs bar and away to safety, keeping the tie at level pegging.
Tynie threatened again when Jason Kerr made a great run down the right wing before squaring a through ball to Mark Shiells to strike at goal, but his effort was saved well by Hibs keeper Christopher Burt.
After the break, Mitchell found himself clear through on goal for Tynecastle, but, just as he was about to strike, the ball took a bobble, allowing Burt to clear the ball of any impending danger.
It was Tynecastle who looked the more likely to score, and Jason Kerr's deflected shot could have given them the luck in front of goal they craved, but his shot drifted wide of the mark.
Substitute Danny Brooks was next to step up for Tynecastle, and the defender attempted a speculative shot from 40 yards out which dipped and swerved, before being gobbled up by Burt who held well.
Edina's Grotlin was next to test Brennan in the Tynie goal and fired in a long, swerving effort of his own, but the keeper was equal to the task.
The breakthrough did come straight after, though, when Tynecastle broke quickly down the left hand side for Lewis Murray to cross in for Connachan to bundle home from five yards and put them one up with only a few minutes remaining.
Edina Hibs' last chance to equalise came almost on the final whistle, when Kai Wilson crossed into Ryan Dawson who took a touch before slamming a shot high and wide over the bar to signal the final whistle and confirm cup victory for Tynecastle. Tynecastle manager Simon Kinghorn praised his team for avenging their cup loss three days earlier and also opposition for creating an entertaining cup tie. He said: "We played well, but on Monday we lost and tonight we won - it's just one of those things. It's a fine line between winning and losing. I think it's a fair reflection to have won one game each."
Edina Hibs manager John Hall also praised the two sides for what was an evenly-matched final, but felt his team never performed to the best of their ability on the day. He said: "To be honest I'm a little bit disappointed in our performance. We certainly didn't get going today. I thought we started the second half quite well, but unfortunately we let in a goal and the better team won on the day"
Tynecastle: Sean Brennan, Angus Luff, Max Michie, Danny Brooks, David Wright, Craig Russell, Jaimie Purves, Robbie Ewing, Callum Sutherland, Jason Kerr, Greg Binnie, Lewis Murray, Mark Shiells, Kyle Mitchell, Tyler Connachan, Greg Kerr.
Edina Hibs: Christopher Burt, Colin Parrott, Aaron Waugh, Jack Pullar, Alex Fotheringham, Ryan Smith, Cameron Jeffrey, Connor Dyet, Jonathon Grotlin, Jack Wright, Ryan Dawson, William Noble, Kai Wilson, Christopher Scott, Lewis Mackay.