When the visitors failed to clear after one of many Swans attacks, the ball ran out to Rangel who fired crisply through a crowded penalty area and into the far corner of the net.
The Swans made one change to the side that won at Tranmere with Leon Britton replacing an unfit Warren Feeney.
The pace of Paul Anderson and the forceful running of Andy Robinson caused the Hartlepool defence early problems and apart from one or two isolated visiting attacks that came to nothing, it was one-way traffic towards the Pools goal.
Totally against the run of play however, midway through the opening period, James Brown found himself free inside the Swansea half and his curling shot looked goalbound until City keeper Dorus de Vries palmed it onto the crossbar and behind for a fruitless corner.
Gradually the visitors clawed their way back into the game with Brown looking dangerous whenever he found space down the left flank, leaving the Swans reduced to shooting from distance.
A mistake by Godwin Antwi two minutes into the second half gave Jason Scotland an opening, but Jan Budtz went full length to gather.
Scotland had another opportunity on the hour but his shot zipped across goal and out of play.
The Swans looked certain to open the scoring in the 67th minute but despite Robinson, Scotland and Britton all having chances, they failed to trouble Budtz.
Darryl Duffy was sent on five minutes later in place of midfielder Darren Pratley, but it was Hartlepool's Anthony Sweeney who came closest when his lob rebounded to safety from the crossbar with the Swansea keeper beaten.
The home side finally broke the deadlock nine minutes from time with Rangel's shot, but Sweeney almost levelled with time running out only for de Vries to palm his shot behind for a corner.