Southampton youngster Barnes was one of three loanees to join a day before the contest and his composed finish on 61 minutes gave the Glovers a deserved lead.
The ever-impressive Green completed the scoring with a fine effort into the bottom corner to further enhance the club’s reputation as a giant-killer in the tournament.
It is the fourth third tier side Yeovil have seen off this season and they will find out their next opponents in the FA Cup in Monday’s draw.
Barnes was handed a start up front with the returning Francois Zoko while manager Darren Way also granted Barnsley’s Jared Bird, who joined on Thursday, a go from the beginning in the heart of midfield.
The other new recruits, Corey Whelan of Liverpool and Middlesbrough’s Lewis Wing, were named on the bench.
Town’s two other changes saw Green fill in on the right wing while Bevis Mugabi was installed at centre-back.
The Glovers began in positive fashion with new recruit Bird almost getting off to a dream start inside 30 seconds.
Driving into the left-hand side of the area, the 20-year-old, who joined on Thursday, toe poked past goalkeeper Lukas Raeder; Tyrell Robinson was in place to clear from inside the six-yard box.
The other loan starter in Barnes went close to opening his Yeovil account and the scoring on 10 minute, although he didn’t know much about it. Zoko’s curled effort from the edge of the area was palmed on to the Saints frontman but the ball fell wide of the mark.
Jake Gray hacked over and Otis Khan shot straight at Raeder as Way’s men continued on the front foot.
Winger Green was proving a thorn in Bradford’s with his pace and Barnes’ creative movement was causing a nuisance.
One of the latter’s runs was picked out expertly by Zoko and his shot towards the far corner needed to be palmed behind. The resulting corner saw Tom James power through a host of bodies and past Raeder but the ball cannoned off Matt Kilgallon on the line to safety.
Charlie Wyke flicked over in a rare visiting attack before another weaving dribble from Bird ended with Raeder smothering at Barnes’ feet.
For all the hosts’ play, they deserved a goal but it was the Bantams who nearly provided one in first-half stoppage time. Hitting Town on the break, the ball eventually fell to Paul Taylor and his blistering drive warmed Artur Krysiak’s hands ahead of the referee’s half-time whistle.
Boosted by a encouraging end to the first period, Bradford came out of the blocks quick following the interval. Timothee Dieng glided into the box and tested Krysiak from close range, then Shay McCartan found the Thatchers terrace.
Those in the home areas were up cheering when they thought Khan’s free kick nestled into the back of the net rather than the side of it, and Wyke’s low attempt was tipped behind brilliantly by Krysiak at the other end.
The game had become stretched and exciting with some excellent last-ditch interventions before Barnes deservedly scored just past the hour.
Thread through by Khan, the 21-year-old forward raced one-on-one with Raeder and coolly slotted underneath the German stopper.
Barnes’ confidence increased and it showed as he won possession back 30 yards at goal and shot straight at Raeder.
Town did have to be alert to the Bantams’ threat and Taylor sent an attempt crashing against the crossbar and Krysiak superbly denied McCartan on the follow up.
Not only did the Polish keeper’s heroics preserve the Green and Whites’ lead, it allowed them the opportunity to go further ahead.
An incredible mazy dribble from Zoko saw the Ivorian travel 30 yards before teeing up Green to emphatically find the bottom corner.
The buoyant home crowd were in fine voice to help Way’s team see out another famous victory for the club in the FA Cup.
It gave the players on the pitch an extra lift in the closing stages to complete the job on a Bradford team competing for promotion in League One.
Who the Glovers face in the next round remains to be seen but the magic of the cup is still alive and kicking in Somerset.
Yeovil Town: Krysiak, James, Mugabi, N.Smith, Dickson, Green, Gray, Bird (Wing 79), Khan (Santos 90), Zoko, Barnes (Whelan 85).
Bookings: Bird 16.
Not Used: Maddison, Golubickas.
Bradford City: Raeder, Hanson, Kilgallon (Thompson 46), Vincelot, Robinson, Gilliead, Dieng (Devine 70), Law, Taylor, McCartan (Poleon 75), Wyke.
Bookings: Vincelot 31, Robinson 50, Dieng 68.
Not Used: Sykes-Kenworthy, Gibson, Patrick.