Both sides were left to rue a number of bad misses, with Town squandering three one-on-ones, while Burnley saw two close-range headers denied by Richard Wright.
Dos Santos and Luciano Civelli both failed to beat Brian Jensen when sent clean through for Town and the latter miss proved extremely costly.
Burnley broke upfield and Elliott gathered the ball 45 yards out, easily beat Ivan Campo and raced into the box before firing an angled drive beyond Richard Wright.
Richard Wright beat away a fierce Chris Eagles drive, and then blocked a free header from six yards by Elliott, who was picked out by Eagles.
Jon Stead put a free header wide, but then won a flick to put Kevin Lisbie clear but he shot inches wide.
Richard Wright was fortunate to keep out a point-blank Steven Caldwell header, which hit the home keeper on the face from just a couple of yards out.
That good fortune proved vital ten minutes later when Town drew level when Stead's flick was collected by Dos Santos, who cut in and lashed a low drive beyond Jensen from 20 yards.
Both sides went for it as the latter stages closed in and both keepers were needed to save their sides.
Richard Wright saved efforts from Robbie Blake and Chris McCann, while Gareth McAuley cut out a pass from sub Jay Rodriguez, which was destined for an unmarked Blake.
Town had a great chance when Dos Santos failed to make contact with a David Wright cross right in front of goal, while Jensen did well to parry a firm Pablo Counago shot.