Two goals from Jon Stead mean Town progress to the next round of the West Riding County Cup, having been made to work hard for victory at The CNG Stadium.
Falkingham named eight senior players in his line-up as Town looked to progress to the next round.
The visitors started confidently, forcing Joe Cracknell into a good save in the seventh minute.
Jack Emmett nearly opened the scoring in the 13th minute. Sam Jones was played through and his effort was saved by the keeper. The ball fell to Emmett on the edge of the box who attempted to place his effort, hitting the post.
Bradford opened the scoring on the 16th minute. Isiah Bazeley made a powerful run down the right, bursting into the box to fire past Cracknell to give the away side the lead.
However, Town responded perfectly, scoring only a minute after the restart. Stead received the ball on the right side of the box and he fired a powerful strike across goal which was out of the reach of the Bradford keeper.
Town looked to press on following the equaliser, creating two chances within five minutes. First, George Thomson took aim from the edge of the box and then Jones nearly gave Town the lead after Matt Taylor whipped a great ball into the box. Jones’ glancing header narrowly fell wide of the far post.
In the 36th minute, Lloyd Kerry played a fantastic ball with the outside of his boot to send Taylor through down the left. He crossed from the by-line nearly causing a mix up in the Bradford defence.
Bradford finished the half strongly with Town having to put in a couple of good blocks to protect Cracknell’s goal.
Both sides looked to be on the front foot at the start of the second half. A great tackle by last man Harry Clark stopped Bradford breaking free and then at the other end Thomson worked some space to hit with his left foot just wide of the woodwork.
Town were starting to have the better chances as the hour mark hit. Jones came closing after making a decisive run with the ball towards goal, he attempted to lift the ball over the on-rushing keeper but was unable to hit the target.
Bradford could and should have taken the lead in the 68th minute when Lewis Knight failed to convert from only a few yards out following the low cross from the right.
Town made The Avenue pay when they conceded a penalty for a handball in the 70th minute, a result of Emmett's strike. Stead stepped up and scored, shooting low to the keeper’s right.
Falkingham’s men faced few threats for the remainder of the game with Bradford playing the remaining five minutes with ten men after Joe Ockerby was forced to leave the field with injury after all three substitutes were used.
Another superb challenge from young centre back Clark stopped the last attempt for Bradford as Town progressed at their Yorkshire rivals expense.
Town: Cracknell, Fergus, Hartley, Clark, Thomson, Emmett, Jones, Taylor, Stead, Kerry (Prata 76), Brown.
Subs not used: Wilton, Chippendale, , Haswell, Wicks.
Goals: Stead 17, 70 (p)
Bradford Park Avenue: Green, Bazeley (Mayfield 76), Ockerby, Marriott, Hughes, Priestley (Lyons 59), Rathbone, Lund, Sweeney (Knight 59), Hurst, Johnson
Referee: A Bannister
Report by Henry Whitaker