The Gunners striker let fly from the edge of the area after eight minutes to put the visitors in charge before doubling their advantage with another precise finish early in the second half.
Denilson's deflected long-range effort sealed a comfortable evening's work for Arsenal as they maintained their unbeaten start to the season without really breaking sweat.
As expected, manager Arsene Wenger shuffled his pack, with not one member of the side which kicked off Sunday's draw with Liverpool named in the his starting line-up.
Theo Walcott, Nicklas Bendtner and Eduardo all started in attack, Gilberto Silva was handed a place in midfield and Lassana Diarra made only his second start for the Barclays Premier League leaders at right-back.
United boss Bryan Robson made changes of his own, retaining just four of the side which kicked off Saturday's Coca-Cola Championship draw with Hull. There was no place in the squad for leading scorer James Beattie, with Jon Stead and Billy Sharp given the nod up front.
Neither goalkeeper was troubled in the opening seven minutes but a moment of brilliance from Eduardo soon changed that.
Bendtner slid the ball across to his strike partner on the left edge of the area and the Brazilian-born Croatia international hammered an unstoppable drive past Ian Bennett at his near post.
Wenger opted to start with Walcott down the right flank and the 18-year-old's pace and trickery posed plenty of problems for the Blades defence.
He got in behind United left-back Chris Armstrong and found Bendtner who, again unselfishly, squared to Denilson, but the midfielder's Brazilian technique deserted him at the worst possible moment as his shot soared well over the bar.
The home side threatened briefly and Gunners goalkeeper Lukasz Fabianski, perhaps questionably, opted to punch one dangerous cross when catching it appeared the easier option.
With the Pole's handling less than convincing at times, it was no surprise the Blades looked at their most dangerous from set-pieces.
They could count themselves unfortunate when Chris Lucketti's dangerous header across goal from Nick Montgomery's floated free-kick came to nothing.
Robson's side were almost made to pay moments later, however, when Armstrong sold his goalkeeper short with a backpass but Bennett was able to beat the advancing Eduardo to clear to touch.
Again Walcott was the creator in the 34th minute when he slid the ball across the face of goal and only Lucketti's last-ditch challenge averted the danger with Bendtner ready to pounce.
Comparisons have already been made with Gunners legend Thierry Henry but the England international's finishing failed to match the Frenchman's high standards seven minutes from half-time when he timidly dragged a shot from the right side of the area from Abou Diaby's pass.
It was a case of deja vu early in the second half when Eduardo notched his second of the game with another clinical finish after 50 minutes.
The 24-year-old, who has scored 10 goals in as many games in Croatia's European Championship qualifying campaign, raced onto Kieran Gibbs' defence-splitting pass and calmly slotted beyond the onrushing Bennett into the bottom-left corner.
United beat the Gunners on their last visit to Bramall Lane - a 1-0 Premier League victory in December - but Wenger's side are a different proposition these days.
Unsurprisingly, the Championship side spent most of their time chasing shadows as the Gunners moved the ball around at will with some slick interplay.
However, they continued to work to prise an opening and were almost rewarded when Montgomery whipped a teasing cross in from the right but Stead could only divert the ball wide at full stretch.
Walcott's profligacy was again apparent when he rounded Bennett but could only hit the post with the hard work done, but the game was put beyond doubt in the 69th minute when Denilson's 25-yard strike hit United skipper Chris Morgan and looped over the wrong-footed Bennett.
The experience of Gilberto came to the rescue of Wenger's young side midway through the second period when the Brazilian retreated to the Gunners line just in time to hook Lucketti's goalbound effort to safety.
Robson brought on Danny Webber, Alan Quinn and Keith Gillespie as his side began to tire but the changes had little impact.
Eduardo was denied his hat-trick late on when Morgan managed to divert another well-struck shot for a corner and he was forced to settle for a brace after being replaced by Nacer Barazite in the 85th minute.
With the game wrapped up, Wenger's side were happy to soak up the pressure and claim their first victory over United at Bramall Lane inside 90 minutes since 1991.