A freak Neil Austin own goal early in the second half gave Nigel Worthington's side the advantage and as Barnsley went hell for leather for a leveller Logan produced a string of vital stops.
With Championship survival the prize for a win by either side and a point apiece possibly enough for both Oakwell was crackling with nervous tension even before the start.
Barnsley marked the last home game by slashing admission prices to £10 for adults and £5 for youngsters, but it backfired when they were swamped with fans and hundreds were locked out.
All they missed in the first half was a dreary stalemate as both sides were cautious and twitchy.
Four minutes into the second half Leicester struck an triggered a frantic second period.
A long clearance from the back bounced over the home defence and Matty Fryatt was quick to react and lobbed the ball over Nick Colgan as the keeper raced off his line. Austin tried to head the ball out with Fryatt on his heels but could only steer it into the open net.
Two saves in two minutes preserved the advantage for the home side when Barnsley's Danny Nardiello - with six goals in the last four matches - twice burst into the area and saw two rasping shots superbly turned away by Logan.
And Logan completed an impressive hat-trick of saves when he dropped and made a brave stop at the feet of Brian Howard after he raced at goal after a slick link with Martin Devaney.
Then Devaney dazzled down the left and drilled an inviting ball across the face of goal but Nardiello was a fraction away from a scoring touch.
Devaney created a chance for himself as the minutes ticked away but after he cut in from the left and smashed a long range shot wide.
But both sets of fans celebrated together as news came through that Barnsley were safe.