Hudson-Odoi limped off helped by two members of Chelsea's medical staff after sustaining the injury in the first half of the Premier League clash at Stamford Bridge.
"Really gutted to end my season with an Ruptured Achilles, gotta work hard and try and come back stronger for next season!!!" Hudson-Odoi tweeted.
Chelsea assistant manager Gianfranco Zola admitted he feared the worst when he faced the media before Hudson-Odoi's tweet.
"Hudson: it looks a serious thing. Before I can tell you how serious, we have to wait until tomorrow. But it's not looking good. It's a problem with his Achilles tendon," Zola said.
The 18-year-old's injury is a major blow for Chelsea as they battle to qualify for next season's Champions League either by finishing in the Premier League's top four or winning the Europa League.
They are fourth in the top-flight and face Eintracht Frankfurt in the Europa League semi-finals.
Hudson-Odoi has also broken into the England team this year and his injury means he is likely to miss the Nations League semi-finals in June.
His injury completes a rollercoaster season which saw Hudson-Odoi strongly linked with a move to Bayern Munich after growing frustrated at his lack of first-team action with Chelsea.
He eventually convinced Chelsea manager Maurizio Sarri to start him in the Premier League towards the end of the season following a pair of impressive displays for England.
Hudson-Odoi's fine performances for Chelsea in recent weeks were one of the few bright spots in a turbulent campaign.
However, Zola expects France midfielder N'Golo Kante to recover quickly from a "contusion" to his ribs that forced him to come off at half-time against Burnley.