Del Frisco's, which owns Eagle Steak House and Frisco's Grille chains, said its shareholders will receive $8 per share, which represents a premium of 18.9% to Friday's close.
Including debt, the deal is valued at $650 million. As of March 26, the company had $336.1 million in debt.
Del Frisco's shares were up 17.4% before the bell at $7.90, slightly below Catterton's offer price.
The restaurant chain in December said it was evaluating options for its business, including a possible sale, amid pressure from activist investor Engaged Capital, the company's third-largest shareholder with a near 10% stake.
Engaged Capital pushed Del Frisco's to add a board member, saying the company's steakhouse restaurants were poorly managed and had rushed into buying two chains to avoid an acquisition.
Del Frisco's said on Monday Engaged has agreed to vote in favour of the acquisition, which is expected to be completed by the fourth quarter of 2019.
The company bought two restaurants - Barcelona and bartaco - for $325 million in cash last year, even as traffic declined in its own restaurants.
Upon closing of the deal, L Catterton said it plans to run the bartaco and Barcelona businesses separately from the steakhouse brands.
L Catterton has also invested in restaurant concepts such as Bloomin' Brands Inc, Unclue Julio's Mexican restaurants, P.F. Chang's and Cheddar's Scratch Kitchen.
Piper Jaffray & Co was the financial adviser to Del Frisco's, while Credit Suisse served as financial adviser to L Catterton.
(Reporting by Nivedita Balu in Bengaluru; Editing by Shailesh Kuber)