James Beattie ended a seven-game goal drought with a first-half header to put his side in front in the Championship clash, but Lee struck nine minutes into the second half to earn the home side a point.
The point, however, was not enough for the Tractor Boys - who announced before kick-off that their move for Portsmouth striker David Nugent was off - to keep their place in the top six, as Plymouth Argyle overtook them with their win over Colchester.
Ipswich named former Blade Alan Quinn among the substitutes but did make one change as Danny Haynes started in place of Tommy Miller - while the visitors had to replace the suspended Stephen Quinn and did so with Gary Naysmith.
The home side made a bright start, and David Norris and Velice Sumulikoski both tested Paddy Kenny inside the first 10 minutes.
The Blades, who began the game without a win in their last six league matches, had their first sight of goal in the 19th minute when Beattie met David Cotterill's cross - but the former England striker headed just wide of the target.
It was the home side who continued to have the better of the clash, and they went close again when a Norris corner was cleared only as far as Sumulikoski - only for the Macedonian's shot to fly just past the post.
However, the first goal came at the other end when Michael Tonge crossed from the right - and Beattie headed home from close range.
Haynes had a shot blocked by Chris Morgan as the home side tried to find an instant response, but the Blades held firm to take their lead into the interval.
The Blades, who began the game without a win in their last six league matches, were straight on the front foot in the second half and were inches away from doubling their lead when Tonge's shot from the edge of the box thumped the crossbar.
But the home side were back level after 54 minutes when Jon Walters crossed and Lee sent a superb header past Kenny from 12 yards out.
Haynes twice fired wide from way out as Town looked to find a second, but both sides were finding opportunities at goal hard to come by.
Norris did have one shot blocked before Quinn came off the bench and saw a free-kick blocked by the wall, before Stephen Bywater made his first save of note from another well-struck Tonge effort.
David Wright fired wide for the home side before Tonge's free-kick hit the wall and the rebound from Gary Speed was comfortable for Bywater.
It was the Blades who finished the strongest and Hulse had a powerful header saved by Bywater and Chris Armstrong's shot was blocked by Richard Naylor, but there was to be no winner.
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