Beattie struck first from close range on 32 minutes to put the Blades in front, but Lee replied eight minutes after the interval and overall a draw was probably a fair result on the night.
Ipswich began well with David Norris forcing an early save out of Paddy Kenny, who then dropped smartly to hold a curling free-kick from Velice Sumulikoski on his goalline.
Beattie glanced a header wide from David Cotterill's cross on 20 minutes as United threatened for the first time, before Sumulikoski's cross picked out Norris but he blazed over.
United went in front when Dan Harding carelessly conceded possession to Michael Tonge, who danced into the penalty area and crossed to the far post for Beattie to head home with Ipswich appealing in vain for offside.
Town could have been level a couple of minutes later when Pablo Counago got to the byline and crossed for Danny Haynes to fire in a deflected shot which again Kenny held right on the line.
The second half saw both sides hit the woodwork and then Ipswich equalise all in the space of a couple of minutes as the game swung from one end to the other.
First, Tonge sent a thunderous left-foot strike from 20 yards against Stephen Bywater's crossbar, but Ipswich then went up field and won a free-kick which David Wright lobbed into the box, Jason De Vos headed goalwards and Gary Naysmith sliced against his own post as he tried to clear.
United escaped then, but Ipswich were level moments later when Jon Walters escaped Chris Armstrong's challenge close to the corner flag and crossed for Lee to thump his header beyond Kenny.
Referee Kevin Friend deserves a pat on the back for playing a good advantage after his assistant had signalled for a foul against Walters.
Tonge sent a fierce 20-yard drive just wide soon after the goal and Haynes twice fizzed in low shots narrowly wide of Kenny's opposite posts as both sides looked for the important next goal.
Bywater comfortably dealt with another Tonge shot before United had a decent shout for a penalty in stoppage time as Bywater and Richard Naylor got in a muddle, the striker nipped in and was then knocked over, but referee Mr Friend declined the appeals.