Despite the visitors' dominance in the first half, Juan fired in a 54th-minute free-kick to put Ipswich on course for victory until Webber tucked home a late equaliser.
Sheffield United were hoping to put behind them the disappointment of their FA Cup exit against Colchester.
Alan Lee made his debut for Ipswich following his switch from Cardiff.
For the Blades, new signing Garry Flitcroft had to make do with a place on the bench after crossing the Pennines from Blackburn this week.
It was a slow start for both sides and it was not until the 23rd minute that the first shot on goal was produced.
Chris Armstrong shot wide, before Paul Ifill's effort was cleared by Jason De Vos as the second-placed Blades began to take a grip on the game.
In the 33rd minute, David Unsworth hit a strong left-footed shot from 25 yards which Ipswich goalkeeper Shane Supple did well to parry to safety.
Ipswich mustered their first real chance when Fabian Wilnis fired wide from distance two minutes later.
Ian Westlake's 43rd minute overhead kick produced the first save of the half for Blades goalkeeper Paddy Kenny - but neither side was able to find a breakthrough before the referee blew for half-time.
For all of the visitors' effort, they fell behind nine minutes into the second period against the run of play.
Phil Jagielka gave away a free-kick after fouling Westlake, and Juan stepped up to fire the ball in off the underside of the Blades' bar.
They could have made it 2-0 just after the hour when Danny Haynes squandered a good chance by volleying just wide.
Ipswich were beginning to dominate. But the Blades were still creating chances, and Armstrong headed over from 12 yards in the 66th minute.
United hauled themselves level when Michael Tonge produced a great ball to put Webber through on goal - and the former Watford man made no mistake, placing his shot beyond Supple.
Jagielka's volley with less than five minutes remaining was blocked by De Vos as Neil Warnock's men went looking to snatch a winner.
Steve Kabba could have secured all three points for the Blades deep into stoppage time but shot wide.