The flying winger put City ahead on 25 minutes when he latched on to Steve Howard's flick-on and then beat full-back Terrell Forbes before firing an angled low shot into the far corner.
The first chance had fallen to City after three minutes when Bruno Berner touched a free-kick to Matt Oakley who blasted wide from 25 yards.
Matty Fryatt then turned and drove just wide after Howard had nodded on Paul Henderson's long punt.
Mark Edworthy came to the home side's rescue when a quick break saw Yeovil work the ball to the feet of Danny Schofield eight yards out from goal, but the City full-back made a timely tackle before the Glovers midfielder could muster an attempt on goal.
Josh Wagenaar then needed two attempts to smother an Oakley 25-yard thunderbolt with the keeper in action again soon after to scramble away Howard's header.
Dyer then struck to give City the lead, but Yeovil almost hit back immediately when an unmarked Aidan Downes headed Schofield's cross to the far post into the side netting.
The Foxes spurned a great chance to double their lead on 37 minutes when Oakley skilfully brought down Andy King's incisive pass, but then shot wide from 14 yards with just the keeper to beat.
Midway through a slower-paced second-half Henderson was well positioned to cope with a Gary Roberts free kick from 35 yards.
Wagenaar then kept City at bay on 70 minutes when he parried Fryatt's 18-yard drive, with the keeper recovering quickly to push away Dyer's follow-up shot.
Leicester needed the cushion of a second goal to settle their nerves and it almost arrived on 78 minutes when Fryatt flashed a 25-yard free-kick just wide.