Chris Wilder's men continued to threaten throughout the second half of Saturday's encounter in West London, and Kurt Zouma's own goal from a Robinson cross was just rewards for a hard-working performance from the newly-promoted outfit.
Having impressed in the final third against another promoted side in Norwich City last weekend, Lampard would have expected more of the same from the opening whistle against the Blades, but it did not necessarily work out that way.
While the Blues had their moments, the visitors were also showing that they had no intention of sitting on the edge of their own area, and a wayward Luke Freeman strike represented the first effort of the contest.
However, United's hard work was undone in the 19th minute. Goalkeeper Dean Henderson failed to deal with a tame Abraham header under pressure from Christian Pulisic, and it gave the academy graduate the chance to fire through a pack of players and into an empty net.
United deserved credit for remaining unflustered and although Mateo Kovacic went close for Chelsea from an acute angle, the away side should have equalised just before the break with Robinson heading wide at the back post after being found in space by Chris Basham.
Two minutes later, Chelsea and Abraham made their opponents pay. Two defenders got in each other's way when attempting to deal with a long Jorginho pass, and it gave Abraham the chance to take a touch before firing a half-volley into the bottom corner from 12 yards out.
Holding a two-goal lead, Lampard would have been hopeful of a comfortable second half for his side, but any idea of that went out of the window within a minute of the restart.
Enda Stevens got down the left before producing an excellent low cutback for Robinson, who held his nerve to send a first-time effort into the far corner from eight yards out.
Chelsea took their time before being prepared to go back onto the front foot, but Abraham almost completed his hat-trick just before the hour mark with Henderson making a smart stop from his 10-yard effort.
Ross Barkley also saw a low long-range drive kept out by the Manchester United loanee, although the visitors continued to keep the home side honest on the counter-attack.
That pattern of play soon switched around with Chelsea being content to play on the break, and it only encouraged United who got shots from distance on target through Oliver Norwood and Stevens.
As the game progressed towards the full-time whistle, United ensured that Chelsea had to work hard at the back, and that pressure eventually paid off with one minute of normal time remaining.
Robinson's inswinging cross from the left was intended for substitute for Lys Mousset but Zouma got the final touch to the ball, slicing it into the far corner of the net with Arrizabalaga unable to react to palm the ball away.
While the majority of Chelsea's fans accept that their side are going through a transitional period, there were grumbles at the full-time whistle after a second home match without maximum points.
CHELSEA (4-2-3-1): Arrizabalaga; Azpilicueta, Zouma, Tomori, Palmieri; Jorginho, Kovacic (Batshuayi 83'); Pulisic, Barkley (Willian 60'), Mount; Abraham (Gilmour 84')
SHEFFIELD UNITED (3-5-2): Henderson; Basham (Osborn 84'), Egan, O'Connell; Baldock, Freeman (Mousset 79'), Norwood, Lundstram, Stevens; McBurnie (McGoldrick 63'), Robinson