Neil Collins slammed in a third for the visitors late on after Danny Livesey's header had given Carlisle fleeting hopes of a revival.
Once again Carlisle's defensive problems, which had seen them lose their last three home games, resurfaced in the 16th minute as Blackman's cross from the right was allowed to reach Flynn at the far post, leaving him the simple task of prodding the ball past goalkeeper Adam Collins.
Blades increased their lead in the 34th minute when Blackman skipped past Bradley Potts and squeezed his shot beyond Collin from the tightest of angles.
Twice in the space of a minute Carlisle were robbed by the woodwork, first Liam Noble's lob from the edge of the area hit the crossbar then Dave Symington's raking 25-yarder bounced back into play off the post.
Carlisle got their reward in the 71st minute when Livesey's header from a Jon-Paul McGovern corner found the net from eight yards.
But the Blades went 3-1 ahead in the 84th minute when Michael Doyle's goalbound effort was finished off by Collins right under the crossbar.