Strauss, 28, will become eligible to play for Scotland in September next year, after completing the required three-year residency for eligibility.
This could see him play for Scotland at next year’s Rugby World Cup in England and Wales.
Strauss arrived in Glasgow in September 2012 and has made 56 appearances for the PRO12 side. He has scored six tries and has captained the team.
Strauss, who led the Lions to Currie Cup glory in 2011, told Glasgow’s official website: “I’ve learnt a lot working with the players and coaches at Scotstoun and it was a good decision to move to Glasgow from South Africa.
“I’ve enjoyed the last three years at the club and I really like the group of players that are here. I’ve developed playing in the PRO12 and European Rugby Champions Cup and I’m happy to be staying. I’m ambitious and I want to win trophies and I believe I have the best chance of doing that at the Warriors.”
Glasgow also announced that Fijian lock Leone Nakarawa has extended his stay at the club.
Both Strauss and Nakarawa will remain Glasgow players until at least May 2017.
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