After losing their last two league games, Salop will be boosted by the result, but Cheltenham are now without a win in seven league games.
There was little in the way of clear chances in the opening moments, with both sides happy to size each other up.
It did get lively in the sixth minute when the visitors looked certain to score during a goalmouth scramble, only for Jayson Leutwiler to dive in to avert the threat.
Cohesive play from both sides deteriorated as the rain began to fall at the New Meadow, with neither team managing to string passes together on the slippery surface.
The opening minutes of the second half saw more action than the entire first half as three goals were scored in quick succession.
First it was the visitors, when Byron Harrison was the spearhead of a nice breakaway to open the scoring.
Shrewsbury were back level just three minutes later, however, as James Collins managed to find space to curl a fine effort past Carson.
Five minutes later, Shrewsbury took the lead when fine work from James Wesolowski allowed Mangan to slot the ball home.
Shrewsbury earned themselves a relaxing final 20 minutes with a third goal as Nathaniel Knight-Percival did well to head home from a corner.