Carter, signed for £1.5million by Robson when he was in charge at West Brom, broke the deadlock in the 36th minute with a free-kick from 30 yards that foxed everyone including keeper Paddy Kenny as it curled into the far corner.
Robson had wielded the axe, making five changes from the side beaten by Bristol City two weeks ago as Stephen Quinn, playing alongside brother Alan in midfield, almost carved out an early opener, his teasing corner to the far post was met by Beattie whose glancing header from close range flew inches over the top.
Preston, looking for their first away win of the season, caught Sheffield on the break in the 20th minute, Karl Hawley picking out Carter, whose left-foot strike from 12 yards flashed over the bar.
Loan signing Phil Bardsley picked out Beattie minutes later, who was off-target with another close-range header before Preston snatched the lead.
Midfielder Chris Armstrong brought down Billy Jones on the right flank and Carter pumped in a free-kick from 30 yards, the ball evading a packed goalmouth and keeper Kenny as it curled into the far corner.
Robson brought on Luton Shelton and Danny Webber for Jon Stead and Stephen Quinn at the re-start to inject some pace into his side. Webber had an immediate impact setting up Alan Quinn, whose low strike took a deflection.
Keith Gillespie almost caught out Andrew Lonergan with a cross from the right flank, the Preston keeper scuttling back to scoop the ball from under the bar, before Bardsley drilled in a shot from 35 yards the ball hitting the diving Lonergan.
United were back on level terms in the 78th minute after substitute Sean St Ledger handled in the area and Beattie coolly slammed the spot-kick into the left-hand corner.