Bradford looked to be heading for their seventh defeat in nine matches until Weir-Daley bundled the ball in at the far post with their first serious attack of the second half.
Bournemouth should have been coasting to their fourth successive victory after taking the lead in the 61st minute.
Former England winger Darren Anderton rolled a free-kick to David McGoldrick who scored with a deflected right-foot shot from 15 yards.
Bournemouth had plenty of chances to guarantee a comfortable win but threw away every opportunity.
Josh Walker hit the post after 18 seconds from 20 yards and Bradford goalkeeper Donovan Ricketts turned over a powerful rising drive from McGoldrick.
Josh Gowling also headed wide from a good position and Bournemouth should have wrapped up the game with four minutes remaining.
Walker's pass left substitute Luke Summerfield with the goal at his mercy but his weak shot from 15 yards rolled past the post with only Ricketts to beat.
Bradford mustered little threat in the first half and Bournemouth goalkeeper Neil Moss had little to do.
But Moss went full stretch to tip away a dangerous right-wing cross from Joe Colbeck and got down to keep out a low shot from Moses Ashikodi.
Weir-Daley came on in the 66th minute to replace Ashikodi but was perfectly placed two minutes into time added on to make the best of a header back into the danger area by caretaker-manager David Wetherall.
Anderton, who was behind most of Bournemouth's best moves, had a shot deflected narrowly wide as Bournemouth dominated without being able to add to McGoldrick's fifth goal since arriving on loan from Southampton.