Lukaku, 24, forced Chelsea goalkeeper Thibaut Courtois into a save in the first half but otherwise endured an evening to forget, as the well-marshalled Blues defence snuffed out the visitors' attacking threat.
His club teammate Marcus Rashford came in for criticism from Gary Neville for appearing to close his eyes under pressure from Courtois, which led to the England forward missing a straightforward headed chance early in the first half.
But Sutton feels Lukaku deserves more scrutiny for failing to deliver in a big game yet again – the Belgian has only scored 15 goals in 55 games when facing Chelsea, Liverpool, Arsenal, Manchester City, Tottenham and current club United.
"Romelu Lukaku is one of the most physically imposing forwards in the Premier League," Sutton told the Daily Mail.
"Yet, he was bullied all afternoon by Andreas Christensen – on only his fourth league start for Chelsea – and Cesar Azpilicueta, who is almost half a foot shorter.
"The accusation levelled at Lukaku is that he does not turn up for the big games. I do not like calling players out for a lack of effort, but on this evidence, it is hard to argue."
Sutton also reserved criticism for Rashford for missing a free header under pressure from Courtois.
"When you go up for a header, getting hurt is the last thing you think about," he said. "You only have eyes for the ball – scoring is your sole aim.
"Marcus Rashford is a superb talent who has so many positive attributes.
"I am sure his teammates will have dug him out for ducking out of the header – but did any of them have the right to do so after that performance?"