Torquay, who parted company with manager Leroy Rosenior earlier in the week and who are in the temporary charge of former Shrewsbury and Swansea midfield man John Cornforth, ultimately won a scrappy, untidy encounter with a 64th minute header from ex-Newcastle striker Paul Robinson.
It was a powerful headed goal that earned the visitors three crucial points in their battle to stave off relegation from the Football League.
Shrewsbury boss Gary Peters, who has seen his side play seven matches against League Two's bottom seven teams this term and collect just two points, was at a loss to explain his side's continuing poor showing against the stragglers.
He said: "I watched the lads in training yesterday and they looked as sharp as I have ever seen and we have spoken at length about why we have struggled against teams near the bottom."
The home side, forced into a late reshuffle due to influential midfielder Jamie Tolley suffering with flu, almost made an early breakthrough, but Kelvin Langmead headed wide from close range in the fourth minute.
The next effort of note did not arrive until added time in the first period when Colin McMenamin's low drive drew a diving save from United keeper Andy Marriott.
On the hour Langmead went even closer but this time saw his header rebound from the crossbar, but four minutes later the issue was decided as Robinson rose unmarked to nod home a fine cross from Martin Phillips.
In the 74th minute Langmead had Shrewsbury's best opportunity to level, but saw his lob beaten away by Marriott and the Torquay rearguard then held firm to secure a vital win.