Gary Johnson's side overturned a one-goal first-leg deficit to seal a place at Wembley, holding off long periods of Blades pressure to reach a second end-of-season showdown
The Glovers got off to the perfect start after just six minutes when Dawson latched on to a fine through-ball from James Hayter to take the lead and level the aggregate score.
United struggled to get in to the game, with their first effort on goal coming after 16 minutes, but Kevin McDonald's attempt from 20 yards sailed off target and out of the stadium.
The Blades continued to press as the first half drew on, however, with Michael Doyle and Jamie Murphy going close, before Joe Edwards tested goalkeeper George Long from the angle up the other end.
And Murphy again proved a threat as half-time approached with a fine curling shot from the edge of the area that clattered against the crossbar with Marek Stech beaten all ends up, but the visitors failed to produce before the break.
United started the second half as they they finished the first as Chris Porter and Joe Ironside both drove narrowly wide when in good positions, but Yeovil could, and perhaps should, have extended their lead when Paddy Madden connected with a Luke Ayling cross six yards out but failed to hit the target.
Dave Kitson was introduced with 20 minutes of normal time remaining to offer United a different attacking threat, but it was Yeovil who continued to offer more danger in the final third and again missed a fine chance to seal their progression to Wembley as the clock ran down.
Hayter released Sam Foley on the left, with the 26-year-old's whipped cross returning the favour inside the area, but the striker failed to keep his header down.
But it was third time lucky for the Glovers as, with less than five minutes on the clock, Upson was picked out by Paddy Madden's inswinging cross to head home from close range and give Yeovil the aggregate lead for the first time in the tie.
And there was still time for Yeovil to nearly add a third as the game entered injury time with Madden again, this time finding Foley but the 26-year-old fired wide from 12 yards.