Bradford 3 Northampton 3 City 5-3 pens
Clive Platt's spot kick gave the visitors the advantage following Higgs' save from Wells but Gary Jones, Will Atkinson and Stephen Darby capitalised on unsuccessful attempts from Danny East and Kelvin Langmead to leave Ricky Ravenhill with the task of securing City's place in the 2nd round with an emphatic finish.
With all seeming to be lost on City's behalf Stephen Darby pumped a hopeful long throw into the area; Carl McHugh headed the ball into the air and he latched onto his own flick to loop the ball over Shane Higgs with a header to send the tie to penalties.
Into Extra-Time and after James Hanson was guilty of missing a gilt-edged chance when he fired wide from inside the area, City had a goal ruled out for offside. Gary Jones' delivery was glanced by Rory McArdle and the ball was eventually bundled in but the celebrations were cut short by the linesman's flag.
With seconds remaining Aidy Boothroyd's men equalised to send the tie to Extra-Time; Adebayo Akinfenwa held the ball up well to square towards Platt, McHugh slipped as he looked to intervene and Platt slotted the ball past McLaughlin.
In the last minute Baker's speculative shot was blocked by the arm of Widdowson and City were awarded a penalty kick. Wells stepped up to confidently fire the ball into the bottom corner.
Adam Baker was close to grabbing a late winner when he headed Gary Jones' corner towards goal but the ball was cleared of the line by a player a among the sea of green shirts.
Emyr Huws was the next to try his luck when he received a cross field pass to charge into the area but he lashed his shot inches over the bar when he had looked likely to score.
On the hour mark the impressive Ishmel Demontagnac was inches away from putting his side ahead; Joe Widdowson's long ball was chested into his path by Clive Platt and he hit a stinging half volley narrowly over.
Northampton began the second half as they did the first and after Scott Brown had given the ball away in the middle of the park they broke quickly; Emyr Huws found himself in acres of space in the area, but he hit his shot horribly wide.
After Adam Baker should've been awarded a penalty when Joe Widdowson fouled him in the area, Northampton were awarded a penalty of their own just before the break. Demontagnac was racing into the area when Ravenhill was penalised for impeding him; Demontagnac himself tucked away the spot kick neatly.
On 35 minutes The Bantams took the lead; James Hanson received the ball on the edge of the area and looked to shape up for a shot but instead he found the run of Will Atkinson who fired neatly past Higgs.
Alan Connell produced City's most notable effort of the game so far on about the half hour mark when he played a neat one-two with Hanson, only to curl his 25 yard shot narrowly wide.
A slight slip from Rory McArdle allowed Demontagnac to be temporarily through on goal; Jon McLaughlin was on hand to thwart him and McArdle cleared the ball away to safety.
The Cobblers carried on their promising start and the ball was worked wide to Wilson who teased Meredith before producing a cross that Clive Platt headed narrowly over.
Another chance came when City only half cleared Ben Tozer's long throw; Tozer crossed the ball back into the area and Lewis Wilson volleyed narrowly wide of the near post when he stole in unmarked.
Northampton started the brighter of the two teams and a long ball forward had McArdle struggling and Ishmel Demontagnac nipped in ahead if him to stab the ball narrowly over the bar from inside the area.
. City, for the third time this season, came out on top in a cup tie after a penalty shoot-out. After 6 goals in 120 minutes of play, The Bantams confidently confirmed their place in the FA Cup Second round tie at home to Brentford