Swede Peter Hanson, a Ryder Cup hopeful, and Argentine Tano Goya both shot 67 to sit tied for second.
Kapur, seeking his first win on the European Tour, produced six birdies in a row and eight in total at the Prosper Golf Resort in the northeastern Czech town of Celadna.
“I was 6 under through my first seven holes so it would have been nice to have added some birdies on the back nine. But golfers are never happy,” he said.
Following a near perfect start, Kapur had two consecutive bogeys on the 10th and 11th holes before fighting back with birdies on the 16th and 18th.
“I’ve never grown up on courses like this – I did on flat courses – but it suits my game as it is not a course that rewards long hitters,” Kapur said.
The year has so far been a disappointment for Kapur after the way he finished 2009, with consecutive top-10s at the Alfred Dunhill Championship and South African Open, where he was runner-up. This year he’s made it to the weekend only seven times in 19 events, and missed his last three cuts.
Miguel Angel Jimenez of Spain, who co-designed the 7,155-yard course, carded 71.
“It’s not much of an advantage knowing the course as people practice a couple of days and get to know it,” Jimenez said.
Jimenez, who is seeking his third victory of the year, hopes for a good result to make the Europe Ryder Cup team.
Next week’s Johnnie Walker Championship at Gleneagles, Scotland is the final event in the race for the nine automatic Europe Ryder Cup spots.
The ?2 million ($2.57 million) tournament returned to the Czech Republic last year for the first time since 1997.
Sweden’s Oskar Henningsson won in Celadna in 2009.