The 19-year-old has impressed since the last pre-season in the summer, making his Premier League debut earlier this term.
After signing, Murphy said: I'm absolutely delighted to sign a new deal here and secure my future with the club.
I've loved every minute since I first signed nearly two years ago. I've been fortunate to work with some brilliant players and coaches - with both the Academy and the first team - and I'm really grateful for the opportunities that have come my way.
It's an exciting time to be a young player at this club - we can all see there is a pathway from the number of players who have made their first-team debuts this season. I just need to keep working hard and hopefully I can build on the progress I've made so far."
Head coach Eddie Howe added: I'm really pleased to see Alex sign a new long-term contract here. It's well- deserved recognition for the work he's put in over the course of the season. He had a great pre-season in America and has gone from strength to strength since, earning himself a Premier League debut and a regular place in our matchday squads.
We rate Alex highly - he applies himself brilliantly every day and he knows if he continues working hard that further opportunities will come his way."