There was a minute's applause at Upton Park on Saturday before West Ham's game against Crystal Palace in honour of the Australian, after which his father and brother came on to the pitch and laid a shirt with his name and squad number, which has been retired, on the centre spot.
Speaking on Goals on Sunday, Allardyce said: "We all felt so much for the parents who have supported Dylan for so long and had so much hope.
"There were so many times that young Dylan looked like he had pulled through and recovered.
"Because he loved football so much, he couldn't wait to get back into training, even though he had been through chemotherapy and wasn't feeling so well.
"Unfortunately it kept coming back and sadly, in the end, the brave fight was no more. It was a great shame for the family and for us, and it was a sad moment when we all found out."
Tombides, who joined West Ham at the age of 15, had turned out for the club's U18 and U21 sides and made his senior debut in the League Cup tie against Wigan in 2012.
"According to all the reports from (Academy boss) Tony Carr, when I first arrived, he was looking like being the one to go straight through into the first team as a teenager," continued Allardyce.
"All the indicators were there that he was going to be that good."
The youngster worked tirelessly to promote awareness of male cancer, supporting the One for the Boys campaign along with celebrities including Hollywood actor Samuel L Jackson, snooker player Jimmy White and Australian singer Peter Andre.
"It is a terrible disease and we have to do as much as we can to fight it," added Allardyce. "We do that in this country through charity events."