Goals from Daniel Mason and Kevin Airlie saw the hosts through after Steven Wright had given Hunterfield a half time lead.
Tynecastle will now meet Cavalry Park in the semi-final.
Referee Thomson declared the pitch playable before kick-off, and both teams found it difficult to create chances in the first half. Hunterfield were the stronger team in the first half and their first clear scoring opportunity came after 18 minutes.
Daniel Lyell was fouled on the edge of the box by Mason after a mazy dribble but Lyell’s free kick went into the wall.
The away team opened the scoring after 40 minutes when Wright’s low shot was well placed into the bottom left corner after Kieran Glen supplied the assist.
It looked like Hunterfield would continue to be on top at the start of the second half. David Liddle shot wide from the edge of the area after Shaun Murray’s shot was blocked in the 49th minute. Two minutes later, Murray could only shoot wide from 12 yards for the Gorebridge side.
Tynecastle gradually began to find their rhythm and they grabbed an equaliser after 59 minutes. Craig Sutherland’s free-kick was not cleared by the Hunterfield defence and Mason powered the ball into the bottom left corner.
They could have scored again three minutes later but Airlie’s volley went over the bar after Ross Gunning’s low cross.
The Hunterfield team still created some chances in the closing stages and would have scored if Glen was able to control a cross when one on one with goalkeeper Danyal Walthew after 78 minutes.
Airlie grabbed the winning goal for Tynecastle with eight minutes remaining, heading into the left side of the goal from Sutherland’s corner kick.
After 88 minutes, Hunterfield substitute Ross Quinn was nearly gifted an equaliser when the Tynecastle defence lost possession but he shot over the bar.
Hunterfield goalkeeper Kyle Hill pulled off a fantastic save in the final seconds of the match, diving low to his right to deny Gary Nicholson. Hunterfield 94 coach Colin Quinn felt that his team were unlucky to lose the match.
He said: “We were probably the better team but we did not perform in the second half. We were a bit unlucky with some of the decisions but we did not threaten their goal enough in the second half. We probably had most of the possession and played the better football but we did not take our chances.
“Our main priority this season is the league. We want to win that and we are top of Section B. We’ve got through to the latter stages of most of the cups and narrowly lost the semi-final of the South East Region Cup. We’re not too disappointed that we’ve lost this cup match, as the league is our priority.”
Tynecastle coach Andy Greig felt that the team was deserving of their win because of their second-half play. He said: “It was a really stuffy match for us in the first half. We didn’t get going and they were on top for most of it.
“Once we got reorganised at half time we were getting into our stride and by the last 20 minutes we were dominating the game. All in all, I think we just about came out deserved winners. We’ve got a really good chance of finishing second in the league and we’re now in two semi-finals and two quarter finals. We have a great opportunity to come away with some silverware.
“The boys have to keep going as there’s still a lot of games to be played. As we saw against Hunterfield, the conditions can sometimes be quite difficult but we have a great group of players here and if we continue to play like we did in the second half I think we will end the season with at least one cup and maybe two.”
Tynecastle: Danyal Walthew, Jordan Gunning, Murray Dobie, Euan Rodgers, Craig Rennie, Gary Nicholson, Fergus Lockhart, Daniel Mason, Kevin Airlie, Ross Gunning, Craig Sutherland, Joseph Greig, Samuel MacNab, Craig Bremner, Mark Hanlon, Stephen Lee.
Hunterfield 94: Kyle Hill, Jed White, Joshua Chisolm, Duncan Wilson, Alistair Elliott, David Liddle, Kieran Glen, Shaun Murray, Daniel Lyell, Steven Wright, Blair Monteith, Ross Quinn.