Goodfellow, 21 last week, netted a brave left-foot shot past the advancing Darren Ward after Bjarni Gudjonsson's cross dissected the Forest defence.
Stoke had taken a 13th minute lead in this high energy clash when Belarussian centre back Sergei Shtaniuk rose to head home an inswinging free kick from Jurgen Vandeurzen.
But Forest, who produced some excellent football, then looked well on the way to the three points which would have lifted them to fourth place in the table.
Paul Hart's men equalised ten minutes before the interval when centre back Michael Dawson headed in a right-wing corner from Eugen Bopp.
Dawson's first senior goal topped another fine display by the highly-rated youngster as he and veteran Des Walker formed a formidable partnership.
Forest moved into top gear with David Johnson giving them the lead with his tenth goal of the season just seconds before the interval.
Bopp again supplied the ammunition, this time with a free kick on the right, and the quick-silver Johnson made no mistake with a well-placed header from the far post.
Stoke made a decisive switch at the interval, replacing Vandeurzen with Goodfellow, who injected some much-needed pace on the left flank.
The speedy youngster soon set up chances for Andy Cooke and Tommy Mooney before he himself breached Forest's defences 18 minutes from time.
Stoke manager Steve Cotterill believed his team had done enough to secure a hat-trick of victories following wins over Brighton and Ipswich in the past week.
"Forest are a very good team but we went at them from the off and only felt we struggled for the last 15 minutes of the first half," he said.
Stoke: Cutler, Thomas, Clarke, Handyside, Shtanuk, Vandeurzen, Gudjonsson, O'Connor, Henry, Mooney, Cooke. Subs: Viander, Hoekstra, Iwelumo, Goodfellow, Marteinsson.
Nottm Forest: Ward, Louis-Jean, Brennan, Dawson, Walker, Williams, Bopp, Prutton, Jess, Johnson, Harewood. Subs: Lester, Roche, Hjelde, Reid, Doig.
Referee: M Pike (Cumbria)