The trade gives the Northeast Division-leading Senators a forward with significant postseason experience. Cullen was a valuable third-line center on the Hurricanes' team that won the 2006 Stanley Cup and reached the Eastern Conference final last year.
"It's a tough day to say goodbye to him," Carolina general manager Jim Rutherford said.
The 32-year-old Cullen has 12 goals and 28 assists in 60 games this season. He is making $2.8 million this year and is scheduled to become an unrestricted free agent after the season.
Picard is a 24-year-old who has four goals and 11 assists in his fifth NHL season and second with Ottawa, and will be a restricted free agent after the season. The Hurricanes also received the Senators' second-round pick in this year's draft.
Rutherford said there were "a handful" of teams expressing interest in Cullen before the NHL was to freeze rosters Friday afternoon for the Olympic break, and the Senators were the most persistent pursuers. Carolina was looking for an NHL-ready player and a second-round draft pick.
Rutherford says Picard "should become a regular part of the Hurricanes' blue line." Picard didn't dress for eight of the nine games between Jan. 18-Feb. 4. He had an assist in nearly 16 minutes of ice time in a 6-5 win against Washington on Thursday night.
"It's been a rough couple of weeks here in Ottawa for me, didn't get to play much, so obviously I was pretty happy to go to a team where they wanted me and I'll get a chance to play, too," Picard said on a conference call.
Earlier this week, Carolina dealt veteran defenseman Niclas Wallin to San Jose for a draft pick.
The Hurricanes matched a season high with their fourth straight win Thursday night — their overtime victory against Buffalo helped Ottawa vault into first in its division — and have won eight of 10 to escape the Eastern Conference cellar.
"It's not probable, but we can still make the playoffs," Rutherford said. "We don't want to quit."