Shane Long looked like he had given the visitors all three points before substitute Bogdanovic was felled by Andy Griffin in the area and he got up to convert the spot-kick.
The Royals dominated for large periods of the game and they will feel aggrieved to have given away a sloppy goal after working so hard to take the lead.
Ivar Ingimarsson and Jobi McAnuff (twice) both saw efforts well saved by Blades goalkeeper Steve Simonsen before Long put them in front nine minutes from time with a calm finish from McAnuff’s through ball.
However, Griffin made a reckless foul on Bogdanovic, who was going away from goal to gift the hosts an equaliser.
The result leaves them two points clear of Championship safety, whilst Brian McDermott’s men stay 12th.
Both sides were made to rue conceding last-minute goals at the weekend, with Kevin Lisbie earning Millwall a 1-1 draw against the Blades, whilst Norwich’s Grant Holt netted to condemn Reading to a 2-1 defeat.
The hosts were looking to end a 10-game winless streak and Adams named five changes to his line-up from Saturday, the most notable of which saw striker Sam Vokes make his debut following a loan move from Wolves.
It was the visitors who had the first sight at goal though with Ingimarsson just managing to get something on Brian Howard’s inswinging free-kick but Simonsen was able to make a fine save.
The Blades shotstopper was called into action again when he made a scrambling stop from McAnuff’s header at the back post.
That served as a wake-up call to the hosts and they enjoyed a good spell of pressure, with Mark Yeates looking particularly threatening.
However, their failure to create any clear-cut opportunities was telling and it nearly came back to haunt them.
Ian Harte missed the target with an unmarked header, whilst Howard’s shot from the edge of the area just missed the target.
The game started to become scrappy as the half wore on, although the Royals were the more lively side.
They continued to dominate proceedings after the interval and Simonsen had to produce a smart save from McAnuff’s 25-yard effort.
On the next attack, the Reading winger went past Stephen Jordan to go clean through on goal, but once again he was denied by Simonsen from a tight angle.
Harte, renowned for his dead-ball expertise, saw a 25-yard free-kick go inches wide of the left-hand post.
The home support was starting to grow restless though they were almost given something to cheer about when Michael Doyle met Bogdanovic’s cross but his looping header went wide.
That miss proved costly as Long gave Reading the lead in the 81st minute.
The striker, on his 100th appearance for Reading, latched on to McAnuff’s through ball before sliding the ball under a helpless Simonsen.
United, to their credit, did not give up hope and Bogdanovic drew a foul from Griffin in the area and he slotted home the resulting penalty seven minutes later to give his side a point.