It was a third League Two victory on the bounce for Keith Millen’s side as they moved six points clear of the relegation zone.
Bradford had the better of the early exchanges and right-back Oscar Threlkeld fired over from Lee Angol’s low cross.
But they lost top-scorer Andy Cook through injury after half an hour.
Gibson shot over the bar for Carlisle before the visitors thought they had grabbed the lead with a Yann Songo’o header, but it was ruled out for a foul on goalkeeper Mark Howard.
Carlisle did go in front in added time at the end of the half, Gibson collecting Jack Armer’s pass and beating his man before firing into the top corner.
Gibson set up Jon Mellish early in the second half but his effort flew into the side-netting.
Carlisle made sure of the win when substitute Patrick held off Fiacre Kelleher to slot home the second.