Goals from Connor McKinnon and Stephen Scott saw the Midlothian team secure a tie against Edinburgh United in the next stage of the competition after Elias Mukora had given Edinburgh South the lead.
The home team went ahead after 24 minutes when Mukora directed his left-footed shot into the bottom right-hand corner of the net from 20 yards out.
Eskmill replied immediately, McKinnon volleying the ball past South goalkeeper Harry Cairns after a Grant Stewart cross.
Reds squandered the chance to go farther ahead when Darren Clegg shot wide when one-on-one with goalkeeper Jack McInnes in the 31st minute and striker Connor Burke headed over the bar three minutes later after a Mukora flick-on.
After gaining some composure, Eskmill took the lead after 38 minutes when Stephen Scott’s spectacular free-kick from about 30 yards out powered into the top left-hand corner of the net.
After the break, both teams launched attack after attack on each other’s goal – Eskmill’s Kyle Scott having a shot well saved by Cairns and Red’s Burke hitting a free-kick which shaved the top of the bar.
Both keepers were being kept busy as the tit-for-tat attacks continued.
However, Eskmill’s McInnes was called into action at crucial times pulling off a double save when blocking Jamie Cowan’s initial shot and then clearing Clegg’s follow-up effort. Shortly after, Mukora got on to the end of a Ryan Martin through-ball but shot over the bar when one-on-one with Cairns.
Reds’ Burke almost got the equaliser with three minutes left but his shot went wide.
Edinburgh South Reds coach Sandy Walker felt that his side’s opponents worked harder to earn the victory and said: “Eskmill were hungrier than us today. We did not seem up for it at all. The midfield was not working how they should and Eskmill deserved the win.
“That’s us out of the League Cup so we just need to fight on in the League. We were a Fourth Division team last year and we’ve jumped up to the Second. If we can finish in the top five we will be quite happy.”
Eskmill coach Stuart Baxter is hoping that the team can get promoted to the First Division, saying: “It went well for us today. We fought well and were first to the ball most of the time. The pitch was not very good, and it was not conducive to passing football, but I thought we did well.
“We hope to achieve First Division status next year. We’re third in the Second at the moment but we have a game in hand. Hopefully, we can finish in the top two and win promotion.”
Edinburgh South Reds: Harry Cairns, Liam O’Malley, Shaun Begbie, Declan Grealis, Ryan Martin, Darren Clegg, Liam Ewens, Elias Mukora, Connor Burke, Bradley Kaye, Taylor Philp, Daryl Craig, Jamie Cowan, Denim Collins.
Eskmill: Jack McInnes, Fraser Cumming, David Wardhaugh, Stephen Scott, Stephen Baxter, Grant Stewart, Kyle Scott, Euan Denholm, Mark Cotter, Andrew Lunbong, Nick Brown, Lawrence Naithsmith, James Burns.