Brassard likely will play for the Charlottetown Islanders of the Quebec Major Junior Hockey League this season.
Tampa Bay also signed 21-year-old forward Joel Vermin to a three-year, entry-level contract. The Lightning selected the Swiss native in the seventh round (No. 186) of the 2013 NHL Draft. Vermin had 13 goals and 35 points in 47 games with Bern of the Swiss A-League last season.
"Curtis is a first-round pick and obviously a player and prospect that we think has a good upside," Senators general manager Bryan Murray told the team's website. "We're trying to get our best players tied up and he is certainly one of the guys of the future in this organization so we're very, very happy to get him signed at this point."
The 17th pick of the draft, Lazar had 38 goals and 61 points in 72 games with the Edmonton Oil Kings of the Western Hockey League last season, and then had nine goals and two assists in 20 playoff games.
General manager Lou Lamoriello confirmed prior to New Jersey's preseason game against the New York Rangers on Monday night that Brunner is joining the team and will be on the ice Tuesday morning.
Brunner, 27, spent five years playing for Zug in his native Switzerland before joining the Detroit Red Wings last season. He finished with 12 goals and 26 points in 44 games, then scored five goals and nine points in 14 Stanley Cup Playoff games.