The Magpies faced an early attack from the visitors as Tijani Reijnders raced into the box but the midfielder's strike was directed wide of the right-hand post despite protests for a penalty following Malick Thiaw's recovering tackle on the Dutchman.
Nick Woltemade saw a header cleared off the line by Nico González after Man City goalkeeper James Trafford failed to collect Will Osula's initial header from Kieran Trippier's teasing corner.
In the 17th minute, Newcastle took the lead through Barnes as the left winger, who netted a brace in November's Premier League win against Man City, raced onto Sandro Tonali's inviting through ball before cooly placing the ball into the top right-hand corner.
Aaron Ramsdale produced a key stop to prevent Savinho from levelling the tie after the Brazilian surged into the box after initially intercepting Anthony Elanga's loose pass but the goalkeeper stood tall to block the winger's shot from a tight angle out for a corner.
However, Newcastle found themselves in the 39th minute through Savinho as Jérémy Doku's fizzing cross from the left flank inadvertently bounced off the South American before the ball trickled across the line before Ramsdale could claw it away.
The Magpies booked their place in the Emirates FA Cup fifth round following a 3-1 comeback win away at ten-man Aston Villa while Man City claimed a 2-0 win against League Two outfit Salford City in round four.
Should the tie finish level after 90 minutes, extra-time will be required on the night.
**Newcastle United:** Aaron Ramsdale, Kieran Trippier (c), Lewis Hall, Sven Botman, Sandro Tonali, Harvey Barnes, Malick Thiaw, Will Osula, Anthony Elanga, Nick Woltemade, Joe Willock.
**Substitutes:** Nick Pope, Joelinton, Yoane Wissa, Anthony Gordon, Tino Livramento, Jacob Murphy, Dan Burn, Alex Murphy, Sean Neave.