In the 29th minute, the ball was cushioned down to Stuart Green just inside the area but his shot was turned around the post by Marlon Beresford.
Ryan France went wide as he tried to flick the ball over Beresford, before Boaz Myhill made a great one-handed save to tip over Morgan's 20-yard shot in the 36th minute.
38th Brown was gifted a shot on goal by 5 but Beresford saved it low down.
Nick Barmby thought he'd scored the opening goal but he was well offside.
Seconds later, Luton also had a goal disallowed, for a much more contentious offside call. Andy Dawson played everyone onside and then actually got a touch on the ball as it was crossed in, and Brkovic's close range finish was ruled out.
Myhill made another great saved to deny Carlos Edwards, then in stoppage time Elliott and Cort both had great chances from headers that went off target.
A minute into the second half, Steve Howard hit a shot against the post, and Myhill clutched a Kevin Nicholls effort out of the air.
Ryan France didn't get enough power on his 55th minute shot as he got on the end of a move through the middle, poking in a low shot that Beresford was able to hold on to.
Myhill saved with his legs two minutes later from Sol Davis's shot as Luton again found an opening.
Luton were denied a clearcut penalty after 72 minutes when Andy Dawson handled in City's area but it was missed by the officials. Myhill then made a save from Chris Coyne moments later.
The goal Luton had been threatening came in the 84th minute. Myhill did well to repel Morgan's shot at the near post, but as it was crossed back in City failed to mark up and Howard had a free shot which he blasted into the back of the net.
The game had been full of chances, but Luton looked by far the better team in the second half, and but for an outstanding performance by Boaz Myhill and some suspect decisions by the referee, they might have won more comfortably.