For much of the night, the teenage loanee had been forced to feed on scraps due to York City being the better side.
But, when it mattered most, Cole junior was in the right place at the right time to finish from close range midway through the second half.
It was a finish reminiscent of his famous father, who was in the Oakwell directors’ box to see an entertaining tie settled in the favour of the Reds as Kane Hemmings added a second goal 11 minutes from time.
For the Minstermen, that meant more frustration in cup football courtesy of a seventh defeat in eight ties since returning to the Football League in 2012.
Nigel Worthington’s side had, however, only themselves to blame with the night illustrating why York, despite being unbeaten after five games, are yet to win in League Two and sit 17th in the table.
At the back, York are as solid an outfit as the basement division is likely to see this term. That much was clear in the 65 minutes that preceded Cole’s opener as Barnsley were afforded just once chance of note, Conor Hourihane having a shot that was deflected wide of Michael Ingham’s post.
In attack, a lively York promised plenty but lacked the killer instinct. John McCombe headed wide from close range, while Keith Lowe should also have done a lot better than shoot wide when presented with a close-range chance after the home defence had been deceived by a corner.
A bit more presence of mind at the vital moment would surely have seen both eminently scoreable chances taken but, as has been the case in the league this term, York were found wanting.
This profligacy continued after the break, meaning when Cole tapped in James Bree’s cross there was only going to be one winner as Hemmings then fired in Barnsley’s second from the edge of the penalty area.
Barnsley: Davies; Brown, Nyatanga, Cranie, Dudgeon; Bree (Berry 82), Abbott, Hourihane, Jennings; Winnall (Hemmings 65), Cole. Unused substitutes: Turnbull, M’Voto, McHale.
York City: Ingham; McCoy, McCombe, Lowe, Ilesanmi; Meikle, Summerfield, Penn (Montrose 75), Straker; Fletcher (Hyde 46), Coulson (Hirst 65). Unused substitutes: Mooney, Winfield.
Referee: P Bankes (Merseyside).