First-half goals from Ryan Flynn and Stefan Scougall saw United end a run of four league games without a win.
Flynn scored after 11 minutes with a low shot from inside the area after getting on the end of a low cross from Scougall.
Scougall added a second goal six minutes later with a 20-yard effort which just evaded the diving Chris Weale in the Shrews goal.
Blades boss Clough said: "It was a hard-working performance and I'm pleased with the clean sheet, certainly.
"To get the two goals early settled us down and we didn't sit back. We made Shrewsbury work very hard to create chances against us.
"There are a lot of tired legs. I decided to go with all of the lads who featured the other night (against Fulham) and they did well. It was a very professional job.
"(Jamie) Murphy and Flynn usually have great legs but even they were tired.
"Sometimes you have to grind them out, be professional and be more concerned about stopping the opposition than creating yourself, especially when you have got a two-goal lead.
"We had that against Bradford the other week and then lost the lead so we didn't want to do that again.
"Being second from bottom, we needed the three points. Now all of a sudden we're fourth from bottom and we've got games in hand on teams above - we've just got to make them count."