Barnsley struck late on through Fabio Jalo to progress in the FA Youth Cup despite a valiant performance from City's young Imps.
Fabio Jalo had the first shot of the evening as he forced Isaac Allan into a good save at his near post.
The Imps threatened at the other end in the 13th minute when Bailey Adamson warmed the palms of Rogan Ravenhill with a low strike.
It was a scrappy affair between the two sides, with chances very much at a premium.
Bola Okewoye did almost squeeze the ball in at the near post on the 30 minute mark but Ravenhill denied him.
In sub-zero temperatures the teams went in at half-time with the game goalless after a tight first period.
After the break Jalo curled an effort towards goal from the edge of the box but Allan dealt with the shot comfortably.
In the 53rd minute the Tykes took the lead when Emmaisa Nzondo slotted into an empty net.
City almost mustered an immediate response when Charlie Parks got on the end of a pinpoint free-kick delivery, but his effort went into the hands of the ‘keeper.
The Imps were then presented with the perfect opportunity to level when Zane Okoro was bundled over in the box, with the referee signalling for a spot-kick.
Bailey Adamson stepped up and rifled a great effort past Ravenhill to equalise.
With ten minutes remaining, Jalo ran across the edge of the area before firing an effort inches away from the top corner.
City almost won it at the death but Ravenhill was out quickly to deny Okoro after Berko played him in behind the Barnsley defence.
With the referee blowing the final whistle at one a-piece, 30 minutes of extra-time would follow.
Allan rescued the Imps from falling behind in the first-half of extra-time as he palmed Nzondo’s effort behind.
Barnsley went back ahead in the 113th minute when Jalo finished from just outside the box with the ball kissing the upright on its way in.
City almost levelled in the last minute but Charlie Parks' header hit the crossbar from a corner.
Barnsley: Rogan Ravenhill, Ziggy Kozluk, Robson Woodcock, Trialist A, Jonathan Bland, Bayley McCann, Kieran Graham, Malachi Tommy-Mbogba, Fabio Jalo, Rafael Monteiro, Emmaisa Nzondo.
Goals: Nzondo (53), Jalo (113).
Imps: Isaac Allan, Charlie Carlisle (Orin Aldridge 117'), Noah Simmons, Denny Oliver (Jersey Lopez 113'), MJ Kamson-Kamara, Charlie Parks, Russell Berko, Louis Nesbitt (Daniel Vanderpuye 119'), Bailey Adamson (Tom Nesbitt 97’), Bola Okewoye, Zane Okoro.
Goals: Adamson (65)