Havas Media is the incumbent on the business, having held the account for six years. The pitch process was run by Ebiquity.
M/SIX has been tasked with "implementing a cross-channel media strategy to generate continuous brand visibility and sales growth".
The company wants to tap into digital media, having previously focused on TV ads.
Lucy Milne, the UK marketing director at Just Eat, said: "M/SIX presented a truly impressive, forward-thinking pitch, demonstrating real acuity in digital and data-driven strategies.
"As two new-age businesses with digital and tech at their hearts, M/SIX feel like the perfect partner to help lead our business and brand in the direction we’re aiming for."
M/SIX will create a "bolder, braver digital strategy" to help raise awareness of the brand and drive sales.
Jess Burley, the chief executive at M/SIX, said: "Just Eat is exactly the kind of brand M/SIX were made to work with.
"We’re like-minded businesses with a similar passion for innovation, exploring new territories, and talking to customers in new ways."