The Exiles remain in the League Two relegation zone, a point behind Hartlepool with four matches to play.
The Welsh club were bottom of the table and 11 points adrift of safety when Flynn took over from Graham Westley in early March.
Yeovil stay 20th, but are now only four points clear of the drop zone.
Demetriou fired in left-footed from 20 yards just before the hour as Newport dominated at Rodney Parade to claim a third consecutive win.
Newport County manager Mike Flynn: "It was a huge effort from the players. I know I keep saying but I won't get bored of it they were absolutely outstanding again today.
"I thought we were by far the better team, we just lacked that final cutting edge in the final third. We should have scored two or three today but luckily I told Mickey Demetriou to take the free-kick and he scored."
Match ends, Newport County 1, Yeovil Town 0.
Second Half ends, Newport County 1, Yeovil Town 0.
Substitution, Newport County. Aaron Williams replaces Alex Samuel.
Attempt missed. Kevin Dawson (Yeovil Town) left footed shot from the right side of the six yard box is just a bit too high.
Corner,  Yeovil Town. Conceded by Dan Butler.
Alex Samuel (Newport County) wins a free kick on the left wing.
Foul by Liam Shephard (Yeovil Town).
Attempt saved. Kevin Dawson (Yeovil Town) right footed shot from outside the box is saved in the centre of the goal.
Liam Shephard (Yeovil Town) wins a free kick in the attacking half.
Foul by Tom Owen-Evans (Newport County).
Substitution, Newport County. Darren Jones replaces Mark Randall.
Corner,  Newport County. Conceded by Liam Shephard.
Attempt blocked. Sean Rigg (Newport County) left footed shot from the left side of the box is blocked.
Attempt missed. Jean-Louis Akpa-Akpro (Yeovil Town) header from the centre of the box is close, but misses to the left.
Kevin Dawson (Yeovil Town) wins a free kick in the attacking half.
Foul by Ryan Bird (Newport County).
Ryan Dickson (Yeovil Town) is shown the yellow card for a bad foul.
Ryan Bird (Newport County) wins a free kick on the right wing.
Foul by Ryan Dickson (Yeovil Town).
Corner,  Newport County. Conceded by Kevin Dawson.
Substitution, Yeovil Town. Alex Lawless replaces Matt Butcher.
Foul by Alex Lacey (Yeovil Town).
Alex Samuel (Newport County) wins a free kick in the attacking half.
Attempt missed. Sean Rigg (Newport County) left footed shot from outside the box is close, but misses to the right.
Attempt missed. Alex Samuel (Newport County) right footed shot from the right side of the six yard box is close, but misses to the right.
Attempt saved. Jean-Louis Akpa-Akpro (Yeovil Town) header from the centre of the box is saved in the centre of the goal.
Corner,  Yeovil Town. Conceded by Mitch Rose.
Substitution, Yeovil Town. Shayon Harrison replaces Omar Sowunmi.
Francois Zoko (Yeovil Town) wins a free kick in the attacking half.
Dangerous play by Sean Rigg (Newport County).
Attempt saved. Alex Lacey (Yeovil Town) right footed shot from outside the box is saved in the bottom right corner.
Substitution, Yeovil Town. Jean-Louis Akpa-Akpro replaces Otis Khan.
Goal!  Newport County 1, Yeovil Town 0. Mickey Demetriou (Newport County) from a free kick with a left footed shot to the bottom left corner.
Ryan Bird (Newport County) wins a free kick in the defensive half.
Foul by Alex Lacey (Yeovil Town).
Attempt missed. Scot Bennett (Newport County) header from the right side of the six yard box is too high.
Corner,  Newport County. Conceded by Liam Shephard.
Foul by Ryan Bird (Newport County).
Otis Khan (Yeovil Town) wins a free kick in the attacking half.
Foul by Alex Samuel (Newport County).
Newport County's recovery under caretaker manager Mike Flynn continued as Mickey Demetriou's second-half free-kick saw them beat Yeovil 1-0.