The visitors registered the first attempt at goal in the second minute as Craig Noone gained possession in the final third and fired over the crossbar.
Cardiff had a golden opportunity to take the lead three minutes later when Eoin Doyle beat the offside trap and raced through on goal, only to be denied by the onrushing Cody Cropper in a one-on-one situation.
Doyle tried his luck again in the 17th minute, giving his marker the slip before being thwarted by an acrobatic save from American keeper Cropper.
Cardiff remained in the driving seat for the remainder of the first half, but were unable to find a way past the in-form Cropper, who pulled off saves to deny first Stuart O'Keefe, and then Noone before the referee brought the half to a close.
MK Dons were quick off the starting blocks in the second period, Cristian Benavente firing a warning shot with a half-volley that bounced narrowly wide in the 46th minute.
However, it was Cardiff who took the lead less than 10 minutes later through Alex Revell, who collected a Noone cross at the far post and smashed the ball across goal into the top corner.
MK Dons remained on top throughout the latter stages and might have wrapped the game up in normal time had Samir Carruthers's vicious half-volley in the 84th minute dipped an inch or two lower.
Their winner was the result of effective team play in the second half of extra time as Carruthers and Hall combined to force Cardiff onto the back foot, allowing Murphy to latch on to a loose ball and fire home a debut goal in the 108th minute, sealing his side's place in round three in the process.