It looked comfortable for Albion for the first half-hour as they took a well warranted early lead.
Heggem's header from Alex Mowatt's left-foot inswinging right-wing corner was the summer signing's first goal for the club.
But Albion were pierced by the Blades twice in quick succession - and winger Harrison Burrows was the supply line each time.
First, his low cross cannoned on off two Albion defenders and into the path of O'Hare, who stabbed home as keeper Alex Palmer failed to react.
Then, when the Blades spread it wide again, Burrows was left with too much room and he was able to guide over an even more dangerous delivery, which Campbell coverted.
It was a sixth goal in eight games for the free-transfer August signing, who was allowed to leave Stoke City at the end of last season.
However, in the deteriorating, rain-lashed second-half conditions as Storm Darragh continued to blow fiercely, Albion enjoyed a huge stroke of luck just past the hour.
Fellows' attempted cross from the right corner of the area deflected wickedly into the ground off Gus Hamer and then looped almost comically over keeper Michael Cooper's reach into the top-left corner.