Cochrane, 22, made 40 appearances and scored three goals on loan at Hearts last season. Manager Robbie Neilson said his arrival is key.
“It’s fantastic to get Alex back in permanently,” he told the club website. “He was a huge part of the success the team enjoyed last season so to get him back in as a Hearts player for three years is great news.
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“His versatility on that left side is hugely important to the way we play and I think he benefitted massively from getting a full season of first-team football under his belt.
“He’s a popular lad in the dressing room and obviously there is great familiarity there so I hope he can pick up from where he left off last season.”
Hearts sporting director Joe Savage said: “Everyone was impressed with Alex’s contribution to the team last season so it made perfect sense to try and bring him back in on a permanent deal.
“He’s still a young player but he’s clearly very talented and has a hunger to improve, and – as we’ve demonstrated time and time again - Hearts is a place for talented players to flourish.
“I have to thank Davie Weir at Brighton and Hove Albion for the part he played in making this move happen, and also our own board here at Hearts because they are backing us at a crucial moment in the club’s history.”