Solskjaer scored 126 goals for United in 366 appearances and nearly half his appearances came as a substitute.
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They almost had something to cheer just before the break but Kirkland held on to Solskjaer's diving header.
Ole Gunnar Solskjaer came on for his first appearance in five months after 71 minutes, but made little impact.
Solskjaer did get the ball in the net, but all he earned was a booking for unsporting behaviour towards Schwarzer.
BBC: SPORT | Football | Premiership | Man Utd 1-1 Middlesbrough
Quinton Fortune, Teddy Sheringham and Ole Gunnar Solskjaer are candidates to stand down after a tough game in the Old Trafford mud.
United brought on Ole Gunnar Solskjaer, but it was Boro who next threatened as Lee Cattermole went agonisingly close with a powerful shot.
BBC: SPORT | Football | Premiership | Man Utd 1-1 Middlesbrough
Ole Gunnar Solskjaer's match and season came to an abrupt halt after a late challenge by Ugo Ehiogu, and he was replaced by Jordi Cruyff.
Solskjaer latched on to a Rooney header and tucked the ball across goal and into the bottom corner to further add to the party atmosphere.
Ole Gunnar Solskjaer flicked a long ball into the path of Giggs, who crossed from the left for Scholes to steer the ball home from eight yards.
The home side looked the more likely to break through and Rooney, Alan Smith and Ole Gunnar Solskjaer all had half chances as West Ham held firm.
And for a manager who expects players to come off the bench and make an impact - Ole Gunnar Solskjaer immediately springs to mind - his decision was perhaps understandable.
Ji-Sung scored with an effort from point-blank range after being the quickest to react after Friedel parried a Ronaldo free-kick and Solskjaer added the finishing touch to the win with his late strike.
Semb's problems could have been more severe as, despite earlier fears, he has been able to include the English-based trio of Eirik Bakke, Ole Gunnar Solskjaer and Vegard Heggem, as well as Rosenborg's Roar Strand.
Of United's other current injury problems, Ferguson said both skipper Gary Neville and veteran front-man Ole Gunnar Solskjaer should be back in training ahead of the first match of the season against Reading on 12 August.
Indeed United could have increased their lead before half time as a cross from Ole Gunnar Solskjaer eluded both Saha and Rossi by inches, then Saha dithered as he was put through by his strike partner.
"I think back to that great quartet of strikers - Dwight Yorke, Andy Cole, Teddy Sheringham and Ole Gunnar Solskjaer - who all had a ruthlessness and concentration that saw them determined to finish off games, " the Old Trafford boss added.
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His headed goal looked to be taking Liverpool to their first FA Cup victory over United since the 1920s - but with two minutes to go Manchester turned the game on its head as Dwight Yorke and substitute Ole Gunnar Solskjaer put Liverpool to the sword.
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