Canon does not tell us how old Fluffy was when he was placed in the castle. He may have still been a puppy when put into the room on the third floor, in anticipation of him growing big enough to guard the Sorcerer's Stone. Hagrid does have a history of raising monstrous beasts in the castle (Aragog). In the same vein, Fluffy may not have been fully grown at the time he was guarding the stone, yet still have been very, very big, but manageable for Hagrid.
Or, perhaps a Shrinking Solution was used, as Snape used on Neville's toad Trevor in Prisoner of Azkaban. Once Fluffy was in place, the antidote could be given, returning Fluffy to his normal size. Only one person (Hagrid, presumably) would be needed to administer a Shrinking Solution and its antidote.
[Snape] trickled a few drops [of Shrinking Solution] down Trevor’s throat.
There was a moment of hushed silence, in which Trevor gulped; then there was a small pop, and Trevor the tadpole was wriggling in Snape’s palm.
The Gryffindors burst into applause. Snape, looking sour, pulled a small bottle from the pocket of his robe, poured a few drops on top of Trevor and he reappeared suddenly, fully grown.
Prisoner of Azkaban - page 97 - Bloomsbury - chapter seven, The Boggart In the Wardrobe