Idiakez curled a superb free-kick into the top corner of Marlon Beresford's goal in the 72nd minute and seemed to have settled what had been an absolute turkey of a game with a rare moment of class.
But Mike Newell's troops didn't give up and Brkovic popped up eight minutes later to scramble home after Lee Camp had palmed out Dean Morgan's low cross-shot. A draw was just about the right result in a match where chances were at a premium.
After a desperate start the game sprang to life in the 13th minute when Idiakez produced a sublime pass over the Luton defence for on-loan striker Danny Graham to run on to.
But Hatters keeper Beresford bravely stole the ball off Graham's toe just when he looked set to pull the trigger.
It took another 15 minutes for Luton to carve out their first chance - and Rowan Vine wasted it by firing Carlos Edwards' clever cut-back badly over.
The second half followed a similar pattern to the first with both teams creating nothing more than half-chances in front of Pride Park's biggest crowd of the season - 26,807.
But on 67 minutes Idiakez unlocked the visitors' defence for Graham again - only to see his team-mate denied by the legs of Beresford.
However, five minutes later the Spanish midfielder showed Graham how it should be done when he bent a 25-yard free-kick around the wall and in off Beresford's left-hand post.
It was Idiakez's sixth goal of the season and one that deserved to win any game.
But Brkovic pounced to take his own tally for the campaign to a dozen and end Luton's run of three successive defeats.