He finished 54 holes at 14-under 202 and is one clear of David Drysdale, who finished with an eagle and two birdies for a round of 6-under-par 66.
Kjeldsen's 62 topped Ignacio Garrido's course record at Real Club de Golf de Sevilla by a single stroke. That 62 matched the 18-hole tournament record originally set by Miguel Angel Jimenez in 1999 at a different venue.
Needless to say, this was also a career best on the European Tour for Kjeldsen.
'Today was fantastic,' said Kjeldsen. 'I came out today with no pressure on myself. I knew I had to play well today. As a round goes, I probably played as good as I ever have.'
Kjeldsen plodded along at 1 under through six holes, but then rattled off seven consecutive birdies. The streak was nearly snapped when Kjeldsen almost sank a 30-foot eagle putt at 13. He tapped in for birdie and needed a birdie at 14 to match the European Tour record for consecutive birdies.
The Dane gave himself a decent look from 25 feet to put his name in the record book. Unfortunately for Kjeldsen, his bid for history was no where close. He missed the birdie putt by four feet and holed the putt for par.
'Seven birdies in a row, how can you do that? It was fantastic. You just go with the flow,' Kjeldsen exclaimed.
Kjeldsen parred the 14th and 15th holes, then faced a decision at the par-5 16th. He hammered a drive down the fairway, then elected to hit a soft 7-iron for his second. Kjeldsen's ball stopped 45 feet past the stick.
He rolled in the long eagle putt to reach 14 under par. Kjeldsen was six clear of his closest competitors and needed a pair of birdies on his way into the clubhouse for a magical 59.
'It's just one of those things where things had been going so well,' said Kjeldsen of his long eagle putt at 16. 'I didn't really expect that. I was just trying to hit a nice putt and it went in again.'
Kjeldsen made a par at the 17th and came up just right of the putting surface with his second at the last. He chipped five feet short, but drained the putt, which turned out to be for sole possession of the lead.
Kjeldsen is a two-time winner on the European Tour, including last year's Volvo Masters. He knows a 62 one day doesn't guarantee a trip to the winner's circle.
'Tomorrow is a new day,' said Kjeldsen. 'I'm going into tomorrow with a good frame of mind. It's going to be exciting. After today, it just shows what a day can do for you.'
When Kjeldsen signed his card, he was well ahead of the field, taking questions about whether victory was a formality. Then Drysdale played his final three holes in 4 under par and will join Kjeldsen in Sunday's final pairing. Drysdale posted a six-under 66 to finish at 13-under-par 203.
Colin Montgomerie, who celebrated his 500th appearance on the European Tour this week, struggled to a 3-over 75 on Saturday. The European Ryder Cup captain fell into a tie for 35th at 2-under 214.