Everton offered little other than a late flurry of chances when the game was dead and buried and the result heaps more pressure on boss Roberto Martinez.
The Goodison Park outfit sit 12th in the Premier League table on just 44 points after a disappointing season.
Lukaku was well shackled by Sunderland defender and man of the match Lamine Kone throughout the game while the centre back also found time to grab a brace at the other end.
The big Belgian has netted 25 goals in all competitions despite the Toffees' struggles and has offered a rare ray of light for fans this season.
Lukaku's performance in three words
"Lukaku kept himself quiet... Jesus Christ," Carragher said on Sky Sports in response to Niall Quinn's praise of Kone's handling of the striker.
"Everton never looked like scoring."
Speculation has been rife that Lukaku could be on his way out of Everton this summer with a host of club's credited with an interest in the former Chelsea star.
According to recent reports in the likes of the Liverpool Echo, the Toffees have been weighing up their own move for his brother Jordan which could persuade him to stay.
But following the final game of the season this weekend his immediate focus will be on the European Championships with Belgium in France.