Nade and Eggert Jonsson had close-range efforts blocked and Aguiar volleyed inches over from 16 yards.
Ibrahim Tall, Kestutis Ivaskevicius and Christian Nade also returned to the starting line-up after Saturday's defeat by Inverness.
Nade missed an excellent chance to extend Hearts' lead when he fired over from six yards after Aguiar's cross.
Nade also fired a low effort against the outside of the post and a Gary Glen drive shaved the crossbar.
Other players, including Hearts striker Christian Nade, then became involved before police and club officials intervened to defuse the situation.
Hearts responded and Nade fired straight at goalkeeper Chris Smith from eight yards after latching on to Christophe Berra's knock down.
Christian Nade sent a header over goalkeeper Yves Ma-Kalambay and against the face of the crossbar during an even first half at Tynecastle.
Christian Nade was stretchered off and Eggert Jonsson was sent off for hauling back Jim O'Brien after 71 minutes to complete Hearts' miserable afternoon.
Hearts could have doubled their lead shortly after the break when Driver crossed for Nade, but the Frenchman's effort was blocked by team-mate Wallace.
The Blades broke the deadlock six minutes before half-time, Tonge powering an unstoppable free-kick high past Robert Green after Anton Ferdinand had fouled Christian Nade.
BBC: SPORT | Football | Premiership | Sheff Utd 3-0 West Ham
Nade received his second yellow card on 73 minutes at Ibrox for kicking the ball away after the referee's whistled had blown for an offside decision.
But Velicka missed a great opportunity when Nade found his strike partner in space on the edge of the box and the Lithuanian blasted his volley wide.
And after the match Laszlo expressed his frustrations about his lack of striking options, with Christian Nade again failing to pose a goalscoring threat for the team.
Christian Nade, looking fitter after recent criticism of his fast-food diet, teed up Aguiar on the edge of the box, but the midfielder dragged his shot wide.
Sheffield United make two changes from the team that lost 3-0 at Aston Villa, with Christian Nade and Stephen Quinn coming in for Chris Armstrong and Colin Kazim-Richards.
Hearts had the first opportunity of the match when Grant Munro fouled Christian Nade on the edge of the box, but Andrew Driver fired the resulting free-kick over the bar.
Hearts' troubles deepened moments later when striker Christian Nade, who has not long returned from injury, was stretchered off after seemingly going down with an ankle injury, Glen being his replacement.
Jefferies, who rejoined Hearts for a second spell as manager in January, has money available to strengthen his squad, having released big earners Laryea Kingston, Jose Goncalves, Michael Stewart and Christian Nade.
The game was being played in a frantic atmosphere, on and off the pitch, and Danny Webber - a half-time substitute for Christian Nade - was agonisingly close to an equaliser for Sheffield United.
That decision was clear-cut, but Hibs strongly disputed the red card that completed Bamba's miserable day, although it looked like the defender was the last man as he tripped the rampaging Nade 25 yards from goal.
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