Weber's deal is worth $575,000. Irwin agreed to a two-way contract worth $575,000 at the NHL level and $150,000 at the American Hockey League level.
Both signings were announced Friday.
Weber, 27, has played with the Vancouver Canucks the last two seasons. He had seven assists in 45 games last season after getting 11 goals and 10 assists in 65 games in 2014-15. He has 22 points and 48 assists in 274 career NHL games with Montreal and Vancouver.
Irwin, 28, played two games with the Boston Bruins last season. He also had five goals and 25 assists in 64 AHL games with the Providence Bruins.