Harvey Barnes struck the crossbar in the first half for Leicester
Brendan Rodgers' side have endured an inconsistent season but keep alive their hopes of lifting silverware come the end of the campaign.
The hosts came closest to breaking the deadlock on the stroke of half-time but lively winger Harvey Barnes rattled the crossbar following a one-two with Kelechi Iheanacho.
PSV should have taken the lead in just the second minute when Mario Gotze evaded the offside trap but Kasper Schmeichel spread himself wide to deny the German World Cup winner.
The two sides meet at the Philips Stadium in the second leg next Thursday for the right to meet Roma or Bodo/Glimt in the last four.
That tie saw Norwegians Bodo/Glimt score in the 89th minute to take a 2-1 advantage into the second leg.
This was only Leicester's second European quarter-final following their Champions League exploits five years ago which ended when they were knocked out by Atletico Madrid.
PSV are the more experienced side in continental competition and won the old European Cup back in 1988.
Roger Schmidt's side are second in the Eredivisie, four points adrift of leaders Ajax, and are playing in their third European competition of the season.
They were knocked out of the Champions League play-offs and then came third in their Europa League group to drop down into this newly formed competition.
And Leicester will have plenty of optimism going into next week's tie knowing PSV have lost nine out of their past 10 quarter-finals in Europe.
After a watchable first half, the contest disappointingly petered out and in the end there were just three shots on target between the sides.
Gotze's early effort, thwarted by Schmeichel, was their only shot on target in the entire contest, while goalkeeper Yvon Mvogo's saves from Timothy Castagne and James Maddison were both comfortable to deal with.
Iheanacho, meanwhile, missed a glorious chance when he ran through on goal but dinked his shot wide, while the Nigerian also shot off target in the second period.
England Under-21 international Noni Madueke blazed over from a promising position for the visitors and Ricardo Pereira was lucky not to concede a penalty when he tripped PSV skipper Cody Gakpo in the box, but the referee waved play on.
The Dutch side sat back and seemed happy to take a draw, with Leicester struggling to create goalscoring opportunities, but the result extended their unbeaten run at home in Europe to 10 games.
Last season's FA Cup winners are out of that competition this term and are currently 10th in the Premier League, leaving the Europa Conference League as their only chance of lifting a trophy this year.
-- Courtesy of BBC Sport