Sunday marked the 20th anniversary of the death of the England World Cup-winning captain, who made over 500 appearances for the Hammers during a 16-year spell.
Poppy, 21, Freddie, 16, and Ava, 13, carried the ball out at the Boleyn Ground, while players and spectators from both clubs took part in a minute's applause.
Home fans displayed a mosaic reading "Moore 6", which was the defender's shirt number at West Ham, whose chairman David Gold joined in the tributes.
"He was our England World Cup-winning captain, but also more personally for all of us here at West Ham, a legend at our club and always will be," said Gold.
"It was great to see so many West Ham fans turn out in what was bitterly cold weather. But they wanted to pay their respects to a man we hold very dear at this club."