The Clarets dominated from start to finish as they looked for their fifth successive victory at home this season, but a mixture of poor finishing and a fantastic City rearguard action saw the spoils shared.
Alexander's penalty miss was his first in Burnley colours, but the home side had more than enough opportunities to claim all three points.
Alexander's powerful first-half effort sailed wide of the post after Tyrone Mears had been fouled by Danny Haynes in the 26th minute.
Jay Rodriguez hit the woodwork after the interval but the Robins nearly stole all three points when Jonathan Stead's effort from 12 yards produced a magnificent save from Brian Jensen.
Wade Elliott headed against the bar in the final minute but Keith Millen's men held on for a draw.
The Clarets started brightly as City goalkeeper David James was called upon to save headers from Jay Rodriguez and Clarke Carlisle.
Jack Cork then went close to breaking the deadlock but the young midfielder's 25-yard strike went narrowly over the bar.
Burnley were given a golden opportunity to take a deserved lead when they were awarded a penalty.
Mears latched on to Chris Eagles' lovely chip and, as he went through on goal, the Jamaican international was hauled down by Haynes in the area.
Referee Scott Mathieson wasted no time in pointing to the spot but Alexander fired wide, his first penalty miss for Burnley from 19 attempts.
Alexander looked to make amends eight minutes later but his free-kick was saved by James at the second attempt just ahead of an advancing Cork.
After the interval, Burnley were indebted to Jensen when he made a point-blank save from Marvin Elliott in the Robins' first clear opportunity on goal.
At the other end, Rodriguez's header left James stranded but the effort was cleared off the line by Christian Ribeiro.
Burnley stepped up the tempo and James was on hand to superbly tip Eagles' thumping 20-yard effort round the post.
From a similar distance, the midfielder tried his luck again and though the effort went past the outstretched James, it was fractionally wide of the right-hand post too.
Chris Iwelumo's close-range strike looked goalbound but Ribeiro once more spared the visitors' blushes with a block on the line.
Burnley were nearly caught out on the counter when Stead unleashed a shot that looked certain to be heading in the bottom corner only for Jensen to push it round the post.