Andy Gray had struck early on his debut to give the Blades the lead but Goater fired home from the spot before half-time after Nicky Shorey had equalised to put the Royals in charge.
Despite intense second-half pressure from the away side, Reading hung on for a 2-1 win to move level on points with their opponents.
United boss Neil Warnock signed Gray in the week from Bradford to help breathe life into the Blades' faltering promotion campaign and the Scottish midfielder delivered after just 10 minutes.
Robert Kozluk launched a long throw forward which was cleverly knocked down by Ashley Ward and Gray did not need a second invitation to smash the ball past Marcus Hahnemann.
Reading continued to press and top-scorer Goater fired wide from 20 yards before James Harper created the chance for a deserved equaliser after 25 minutes.
Harper's surging run was brought to an abrupt end on the edge of the box by Blades defender Mike Whitlow and from the resulting free-kick Shorey fired a left-footed effort into Paddy Kenny's top corner.
Reading completed a stunning turn-around in the 34th minute when Goater fired them in front.
Harper was again the source with another strong run at the Blades defence. This time he made it into the box before he was upended by Whitlow and Goater gleefully blasted the ball home from the spot for his sixth goal in seven games.
Minutes later Phil Jagielka had a golden chance to equalise for United but headed Michael Tonge's free-kick past the upright from close range.
United came out fighting after the break and Andy Parkinson's fierce volley from the edge of the area forced Hahnemann into a flying save.
Tonge was hugely influential as the Blades created a host of chances and the much-prized midfielder curled a free-kick wide before Gray spurned a chance for his second when he nodded Tonge's cross narrowly wide.
Warnock threw on Peter Ndlovu and Paul Peschisolido in place of Parkinson and Carl Robinson as the search for an equaliser intensified and Ndlovu almost had the desired impact but failed to hit the target from Jagielka's cross.
Gray then came closest off all to levelling things up when he headed Tonge's corner against the woodwork before Ward saw a similar effort saved by Hahnemann.
Dave Kitson broke to force a save out of Kenny in the final minutes but a third Reading goal would have been cruel on United.