Johnson, who has also been appointed non-executive chairman of the firm, said that he and the company's private equity owners Equistone plan to double the size of the company.
Johnson said: "I bought a stake in the company because I admire the management team, it's a strong brand and it has massive potential to grow. We plan to double the size of the company over the next few years."
There are 14 Gaucho outlets in the UK and the group also owns the CAU chain of 18 restaurants.
Zeev Godik, chief executive of Gaucho Group, said: "Luke has been at the forefront of some of the most dynamic expansion programmes in hospitality.
"He has overseen over 500 restaurant and pub openings, more than any other individual in the sector, and his record in launching successful initial public offerings (IPO) is second-to-none."
However Johnson, best known for owning Pizza Express, said that talk of a stock market listing for Gaucho was "premature".
Johnson has several interests in the dining sector. He is chairman of Patisserie Holdings and Bread Holdings, owner of Gail's Bakery, and was formerly chairman of Giraffe and Strada.
The size of Johnson's stake was not disclosed. Majority owner Equistone acquired Gaucho in January for a reported £100m from fellow private equity firm Intermediate Capital Group.