Teixeira, on loan at the AMEX Stadium from Liverpool, was carried off on a stretcher after 80 minutes of Albion’s goalless home draw on Tuesday night.
The 22-year-old was taken to Brighton’s Royal Sussex County Hospital where X-rays showed a break to his leg just above the ankle.
“Our thoughts are with Joao – as this is a real blow to him after such a good season for the club – and we all wish him a speedy recovery and return to action,” Brighton boss Chris Hughton told the club’s official website.
“He’s been an important player for the club this season, both before and after I came to the club, and I would like to thank him for his efforts during his time on loan here, and also Liverpool for allowing him to come.”
Teixeira scored six goals in 32 Championship appearances, 27 of them starts, for the Seagulls.