The site, which produces cartons for the frozen and chilled foods markets, has taken delivery of the Bobst Expertcut 106 Per Diecutter, alongside the new Speedmaster.
As part of the £7m investment, a Heidelberg XL105 has also been installed at the company's Hoogerheide site in the Netherlands.
Both presses and the Bobst are 3b format, which Nampak said reflects changes in run lengths in the carton market.
Stephen Campbell, site director at Leeds, said the press has a top speed of 18,000sph.
He said: "It stands out in the printing press market because of this and also because of its quick makeready and colour management system."
The company said it has no current plants to invest in additional kit, although it said it constantly reviews its equipment.
Both of the six-colour presses have coater units, and feature Heidelberg Prinect colour systems that integrate press operation, ink delivery and register control.
Production on the Leeds-based press began at the start of October, while the press at Hoogerheide became fully operational at the end of October. The Bobst will be in full production by early November.
Nampak Cartons is part of the South African-based Nampak Group, and its sectors include plastics, healthcare and folding carton packaging markets.
Nampak Cartons spends 4m on kit for Leeds site
Nampak Cartons has invested 4m of a 7m spend on a six-colour Heidelberg Speedmaster XL105 press and a Bobst diecutter at its folding cartons site in Leeds.