The Tigers looked to have done enough to earn a point after surviving a first-half onslaught from the hosts.
But Brkovic nodded home Steve Robinson's cross late on to spark wild scenes in front of a sell-out 9,500 Kenilworth Road crowd.
Hatters, missing leading goalscorer Steve Howard through suspension, were first to threaten when a neat one-two between Brkovic and Enoch Showunmi on the left allowed the Croat to drill a right-footed shot just wide from 15 yards in the seventh minute.
Four minutes later a Town attack on the right saw a Kevin Foley cross deflect off Damien Delaney and fall to the feet of Robinson, but the Irishman saw his ten-yard shot blocked by the legs of Hull keeper Boaz Myhill.
The hosts continued their upbeat start with another whipped-in Foley cross in the 14th minute that saw Brkovic head just wide from 12 yards.
The one-way traffic continued when a long build up down the right involving Alan Neilson, Foley and Kevin Nicholls resulted in Brkovic firing in a low left-footed drive that crept just past the post from the edge of the area.
Showunmi was the next to take on the shell-shocked visitors' defence when he turned past Robbie Stockdale down the left and shot straight at Myhill from 14 yards.
Luton, playing some breathtaking passing football, went close again with a Robinson header that deflected over the bar off Roland Edge in the 28th minute and then with a 25-yard Rowan Vine shot that bounced wide.
Hull's first attempt of any kind came from a speculative 30-yard Aaron Wilbraham shot that ended up nearer the corner flag than the goal, but the Hatters continued their assault with a fierce Showunmi snap-shot in the 37th minute that was pushed aside by Myhill.
Showunmi was bundled aside by Leon Cort as he went for the rebound, but referee Matt Messias waved away the Town penalty claims and instead awarded a corner.
Hull finally mustered up a meaningful effort five minutes before the break when Stockdale's cross fell to Stuart Green, but Marlon Beresford blocked superbly from six yards.
The second half followed at a much more sedate pace with Stockdale laying the ball off for Green to shoot over from 18 yards in the 58th minute.
Brkovic then whipped in a cross a minute later that saw Delaney superbly block Vine's goalbound shot from eight yards out.
Nicholls fired just over for Luton with 15 minutes left before Brkovic's late winner.