On a bright sunny day at Huish Park, the winger calmly stepped up to despatch the spot-kick after referee Keith Hill pointed to the spot following a foul on Tahvon Campbell in the box.
The result means that manager Darren Way maintains his unbeaten record at home, and Yeovil still haven't conceded at Huish Park since February.
Way made four changes to his side to face the Exiles. Darren Ward was injured in the 0-0 draw with Carlisle, and the Town captain dropped out of the starting eleven alongside Ben Tozer, Ryan Bird and Brandon Goodship.
Liam Shephard had signed on loan before the goalless draw on Good Friday, but the returning Swansea loanee went straight into Darren Way’s starting lineup. The three other replacements were Shaun Jeffers, Tahvon Campbell and Connor Roberts, who made his debut for the Wales Under-21 team off the bench against Romania last week.
Newport started the game in a positive fashion, and Town keeper Krysiak had to be at his best to stop Exiles striker Scott Boden from just inside the six-yard box.
But Yeovil were ahead just before 15 minutes. Campbell was floored in the box by the already-booked Holmes and referee Hill pointed to the spot but didn't show the defender a second yellow. Compton stepped up and calmly slotted home the spot kick to give Town the lead.
Campbell and Newport winger Alex Rodman were both shown yellow cards for an off-the-ball scuffle five minutes before half-time and it was to be the last of the action of the half.
The only change at the interval was for neither Yeovil or Newport; referee Keith Hill came off with injury and had to be replaced by linesman Brett Huxtable.
However, Town manager Way was forced to make a change just five minutes into the second half, as goal scorer Compton hobbled off to be replaced by Dawson.
Newport came close to levelling matters when debutant Souleymane Coulibaly headed a whipped cross onto the post, but even if it had found the top corner the striker's effort was ruled for offside.
Jeffers had to be taken off injured after a collision with goalkeeper Joe Day, and the former Newport forward made way for Town top scorer Bird as Yeovil looked to get an important second goal.
Lenell John Lewis came on for Coulibaly with 25 minutes of the game remaining, and the former Grimsby forward was serenaded onto the pitch with a chant of "his name is a shop" by the travelling Exiles fans.
Nathan Smith was booked by substitute referee Huxtable for a foul on Tommy O'Sullivan and with a free-kick in shooting distance, it was a nervy moment at Huish Park. Fortunately for Town, Mark Byrne's shot went high and wide into the travelling fans behind the goal.
The away side wanted a penalty with ten minutes to go as Rodman went over in the box after a challenge by Campbell, but the referee waved away the appeals and Newport were awarded a corner.
Despite pressure from the away side, man of the match Krysiak was relatively untroubled and the Pole held on for another clean sheet and Yeovil earned another valuable three points as 'we are staying up' rang around Huish Park.
Yeovil: Krysiak- Shephard, Sokolik, Smith, Dickson- Roberts, Laird, Dolan (Tozer 77')- Campbell, Jeffers (Bird 58'), Compton (Dawson 50').
Subs not used: Weale, Lacey, Cornick, Lita.
Booked: Campbell 43', Smith 73'
Newport: Day- Holmes, Donacien (Meechan 87'), Jones, Davies- Rodman, Byrne, O’Sullivan, Elito- Boden, Coulibaly (John-Lewis 65').
Subs not used: Green, Partridge, Owen-Evans, , Barrow, Klukowski.
Booked: Holmes 9', Rodman 43', Donacien 86'