New Year's Eve 1905 was, evidently, one of the first to be celebrated in and around Times Square. From the Dec. 31, 1905 NYT:

On the stroke of 12 to-night the figures "1906" will be flashed from the tower of the Times Building in letters of fire that will be visible several miles away...Election night, with its vast and merry crowds, showed conclusively that the center of up-town activity had moved north to Times Square, and there is no doubt, therefore, that the biggest outpouring in the city's history will see the birth of the new year from the streets and avenues immediately surrounding the Times Building.

The last bit sounds like a Dick Clark sound bite.

In another article we learn that they could party hard in 1905:

The customary New Year's Eve celebration, for which practically all of New York turns out, was pushed more than a day ahead this season, and began practically soon after noon yesterday, when the offices downtown closed for the last time in the old year...The old custom of giving away New Year's wishes in quart bottles was revived with a vengeance. Some years ago the Retail Liquor Dealer's Association, by resolution, put a stop to this form of gifts to patrons, but the saloon keepers objected, insisting that they lost business by adhering to the rule. This year there was no attempt to restrict the gifts.