It looked as though City – winners of the season's first phase – was going to coast to its 16th A-League victory of the season when Stefan Colakovski turned in Adrian Luna's low cross in the ninth minute.
However, Goodwin levelled in some style 14 minutes later, collecting a backheeled pass from Lucas Mauragis before nonchalantly curling a stunning 25-yard strike into the far side of the net.
In doing so, Goodwin – aged 16 years and 215 days – became the club's youngest-ever A-League scorer, and second only to Mohamed Toure (15y, 325d) in the competition's history.
The Jets then completed the turnaround with a minute of regulation time to go, Apostolos Stamatelopoulos – who came on for Goodwin – meeting Mauragis' teasing free-kick delivery with a glancing header that found the bottom-left corner.
Jack Duncan in the Jets' net then had to be alert to keep out Nathaniel Atkinson's close-range effort a few seconds after the restart, thus clinching a result that ensured Craig Deans' men moved off the bottom of the table, with Melbourne Victory taking their place.
City finish top regardless, two points clear of Sydney FC, and now turn their attention towards the semi-finals on June 20, where they await the lowest-ranked winner of the elimination-finals.