The spot-kick was awarded for a handball by Henning Berg but Friedel dived low to claim the England striker's tame shot.
The injury problems for Blackburn had been mounting. Damien Duff, Andy Cole, Dwight Yorke and Matt Jansen were all unavailable, so Graeme Souness changed his tactics with a lone frontman, Egil Ostenstad, employed.
The new system worked a treat as his team took the lead through skipper Garry Flitcroft in the 24th minute.
The young goalkeeper went on to have an excellent game. He tipped two long-range David Thompson efforts round the post and touched the same man's free kick on to the crossbar.
Leeds were missing Harry Kewell through illness and it showed as they lacked the cutting edge necessary to break down a resilient Blackburn defence.
Before the penalty, Friedel had made another magnificent block to deny Nicky Barmby what had seemed a certain equaliser.
Gillespie could have added a second in the final minutes when he dragged an effort wide but it did not matter.