Long before Desperate Dan took over the cover of The Dandy from Korky the Cat in 1984 he made a number of front cover appearances. His earliest cover appearance, as far as I am aware, occurred as early as 1945, on the cover of #308 - the Christmas issue for that year. Although not saying anything he does appear in the Christmas feast in the final panel. Of course, his first actual cover appearance would be on the first Dandy Monster annual, but as far as I'm aware his presence on this issue is his first cover appearance on a weekly comic.
He made a very similar appearance on the Christmas cover the following year.
Anyway, Dan's first cover appearance for The Dandy was #308, but he didn't crossover to The Beano for another five years, appearing on the New Year's issue for 1950.
Dan didn't have a change of artist for this crossover, for both Dan and Biffo are drawn by the great Dudley Watkins. Naturally, this wouldn't be Dan's last appearance on the cover of either The Dandy or The Beano, for he would appear on the cover of both many times over the following years.
Images on this post are taken from past Phil-Comics auctions: