A late goal from George Ray extended the Bluebirds’ unbeaten run to seven games and ensured Alexander’s reign got off to a losing start.
Barely 30 seconds had gone when Dom Telford fired home low following Kian Spence’s smart cross for the visitors.
Minutes later Bradford ought to have levelled when Tyler Smith side-footed straight at goalkeeper Paul Farman from inside the six-yard box.
Just past the half-hour mark, Smith made amends when he tapped home another low cross from Jamie Walker.
Soon after the restart, Barrow midfielder Robbie Gotts was denied by a decent Harry Lewis save.
Bradford replied with Emmanuel Osadebe forcing Farman into a brave point-blank save.
Farman was called into action again to push Alex Gilliead’s low drive around the post.
It was against the run of play when Barrow went back into the lead when defender Ray smashed home after sub Jamie Proctor’s initial effort had been blocked 13 minutes from time.