The visiting side had been on a mini-break in La Manga while Rangers had been battling past Queen of the South in their first play-off skirmish and the men in green and white made a bright start in front of 41,236 at a fervid Ibrox.
However, they could not find the requisite composure in the final third to match their slick build-up play and Clark made them pay just before the half-time whistle with a deft finish.
Hibs again pressed at the start of the second period without reward and the wind went out of their sails as veteran striker Miller rolled back the years with a predatory strike, giving the Light Blues a solid advantage ahead of Saturday’s return leg.
Dave King, cleared by the SFA to officially assume the helm in the Ibrox boardroom, was in attendance as the game began at a furious pace with Liam Craig firing wide with the first shot in anger.
Despite the best efforts of the cultured Haris Vuckic and the galloping Lee Wallace, Rangers struggled to gain a toe hold and home hearts were in mouths when the marauding Dominique Malonga picked out the unmarked Jason Cummings at the back post, but the Hibs man tripped over his own feet and the chance went begging.
A series of corners – they were to have 13 in an increasingly frustrating evening – came to nothing for Hibs before Stuart McCall’s men landed a telling blow just before the break.
Vuckic was the architect with a perfectly weighted pitching wedge of a pass that picked out Richard Foster’s run in behind and the full-back’s first-time cross was deftly flicked home by Clark (44) from five yards.
Vuckic warmed Mark Oxley’s gloves with the first serious attempt of the second half, while Malonga could only head tamely over the top after the ball fell kindly for him six yards out.
The second goal the home fans craved duly arrived just after the hour when Dean Shiels’ shot was blocked and Miller (63) raced onto Wallace’s header to curl a cute finish with the outside of his left boot past Oxley.
There was to be no way back for Hibs – who tellingly had only one shot on target all night – as Malonga blazed over and Cummings saw his shot charged down late on.
The second leg is at Easter Road on Saturday where Hibs will need to conjure up a season’s-best performance if they are to keep their promotion dream alive.