The young Glovers midfielder curled in a superb effort midway through the second period and looked to have won the game for the hosts, but a late penalty from the visitors meant a point apiece.
Goalkeeper Tommy Scott received a red card after the spot-kick was scored for dissent.
Yeovil dominated proceedings on the top pitch at Huish Park with Rogers running the show.
He was denied a 12-yard kick of his own after going down over a Newport leg, and Nestor Shako was thwarted in similar fashion moments later.
Despite being in control, Antonio Falanga’s side failed to grab a goal they deserved and needed Scott to pull off to fine stops to keep the score goalless.
Then a piece of brilliance from Rogers opened the scoring. Jack Hilton-Jones saw his drive saved and Rogers picked up the scraps, delivering a shot of quality into the top corner.
It looked to be enough for Town who were comfortable but the referee awarded a harsh late spot-kick to Newport which was dispatched.
Scott was dismissed for dissent after the equaliser and the junior Green and Whites battled hard with 10 men in the remaining minutes to earn a point.