The 21-year-old striker, who scored four goals on his debut while on loan at Birmingham earlier this term, will be available for Saturday's trip to Millwall.
His loan spell will include the Championship play-offs should the Seagulls - currently four points off sixth with a game in hand - be involved.
Brighton boss Oscar Garcia told the club's website: "We are delighted we have finally secured Jesse until the end of the season.
"He is one of the most exciting young English strikers and has already created a lot of interest following his successful loan spell at Birmingham City earlier this season.
"We worked hard to secure a talent who has already shown he can play at Championship level, and I am very pleased to have added Jesse to our striking options for the rest of the season.
"He has an excellent pedigree, having come through the academy at Old Trafford, and there is no doubt that he has a lot of talent and ability; I am really looking forward to working with him.
"There were many clubs interested in signing him, so we are very grateful to David Moyes and all those at Manchester United, for allowing him to come to us here at Brighton & Hove Albion."