The Toronto-based private equity company says it will pay $4.44 billion at the close of the deal for SIG, which is most known as a juice box manufacturer.
It will also pay up to an additional $217 million if the company meets financial targets in 2015 and 2016.
Onex says the deal is expected to close in the first quarter of next year, pending regulatory approvals.
"SIG's management team has successfully proven its ability to enter and grow in new markets, while maintaining its standard of excellence in existing markets," Onex managing director Nigel Wright said in a statement.
"We look forward to partnering with Rolf Stangl and his team to further build upon SIG's impressive track record and continue its growth."
SIG operates seven production facilities in Europe, South America and in the Asia Pacific and has about 5,200 employees.
Meanwhile, Onex said Monday it has also invested in Advanced Integration Technology, a provider of automation and tooling, maintenance services and aircraft components to the aerospace industry.
Terms of the deal, which is expected to close in December, were not released.
AIT has more than 550 employees and facilities in the United States, Canada, Spain and Sweden.