Correct me if I'm wrong, but it sounds like you're in a larger company at the moment. This might be why it is more difficult to drive change - don't let that discourage you though.
I think that in your case, driving change will be more about increased quality than increased efficiency. Think of business critical code, and find a way to convince management that continuously testing the code is valuable. For me, the most critical code has been security sensitive code. Start small, and once others are on the bandwagon, you can push for even more testing.
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I highly recommend reading Li Haoyi's blog post on how to change a company from a software developer's perspective.
Correct me if I'm wrong, but it sounds like you're in a larger company at the moment. This might be why it is more difficult to drive change - don't let that discourage you though.
I think that in your case, driving change will be more about increased quality than increased efficiency. Think of business critical code, and find a way to convince management that continuously testing the code is valuable. For me, the most critical code has been security sensitive code. Start small, and once others are on the bandwagon, you can push for even more testing.