I have seen many people coming up with different definitions of what 'DevOps' is. So in this case I invite you to define your take on what 'DevOps' mean so that others can give answers from the right angle.
I'll assume that DevOps means testing, integration and automation before release.
Assuming that your managers are old-school, but technically oriented people (they have been developers themselves at some point), convincing them means clear demonstration of benefits of switching to a continuous integration system.
It is safe to assume that technically oriented people are more easily convinced by something that 'works'. This is important, because you will need to convince people that the yield on returns from the amount of energy & frustration put into some additional complexity brought by continuous integration is rightfully justified.
How do you do the above? You take an essential algorithm/portion/routine from the project and start converting it to tested + integrated sum of lower order functions. Next you need to nail a bug/challenge that your old-school colleagues could not tackle.
Surviving in the competition has many many parameters, which I think one should carefully consider:
If your team has additional funding and man power:
If your team has limited-additional resources:
If your team has no extra resources and under pressure by live and ongoing project:
'You don't switch the horse while crossing the stream'