Next season will be Browne's 10th with his native province. The Galway man made his senior debut against Cardiff in April 2007, the same year he represented Ireland Under-20's during their Six Nations Grand Slam success.
The flanker-come-second row, who came through the Connacht Academy system, celebrated his 100th cap in January 2014 when he started against Leinster at the Sportsground. The 28-year-old has racked up 91 league and 29 European games in his professional career with Connacht.
Harris-Wright first joined Connacht in 2013 and over the last three seasons has established himself as a key component in the front row. The 26-year-old hooker has been unlucky with injuries and has remained sidelined since last November when he ruptured an ACL in his knee. The former Ireland U20's RWC winner, who has made 47 appearances for Connacht, has seen significant progress in his rehab and will be looking forward to re-joining full training in the coming weeks.
Commenting on the latest deals, Connacht CEO Willie Ruane said:
"We're delighted that Andrew and Jason will remain an integral part of our senior playing squad going forward.
"There are some very important games still to be played but our recruitment and retention programme is certainly taking shape for next season."
Head Coach Pat Lam added:
"From a rugby perspective, these contract extensions are a great boost to our squad. Jason is a key member of our leadership group and contributes a lot off the field to our environment. He has been faced with two long-term injuries over the past two years but he is the type of person who will come through it stronger. We're looking forward to welcoming him back to training.
"Andrew epitomises our vision of Grassroots to Green shirts. He is a Galway man who played all his rugby in the area - club, school and province. He has a lot of experience as someone who can play in the second or back row, and I'm excited as I believe his best is still to come."