Cole signed a four-year, $18-million deal, and Budaj inked a two-year contract worth $2.3 million on Friday at the start of the free-agent shopping season.
Cole earned $3 million last season with the Hurricanes.
Alex Auld, the Canadiens' backup goalie last season, signed with Ottawa on Friday.
The 32-year-old Cole gives the Canadiens the big winger with a scoring touch they were seeking on the free agent market. He had 26 goals and 26 assists in 82 games last season and has posted 184 goals and 204 assists in 620 NHL games.
Later that season, he broke a vertebra in his neck when hit by Pittsburgh's Brooks Orpik. Cole returned to the ice in Game 6 of the finals as the Hurricanes went on to win the Stanley Cup.
Budaj was 15-21-4 with a 3.20 goals-against average and .895 save percentage in 45 games last season with Colorado. In 242 NHL games over six seasons, Budaj is 101-91-27 with a 2.83 average and a .902 save percentage.