Get into their head. Know your audience. Discover their real concerns and interests, whether that's being included on the next round of promotions or getting the deal closed as soon as possible.
Choose your moment. Be aware of their frame of mind and whether or not they will be readily influenced. Just after a difficult meeting with the board probably isn't the right time. Be ready to seize the good moments when they appear.
Put them in a good mood. Gentle flattery is surprisingly effective. If they feel more favourably about you they will be more open to persuasion.
Do as they do. By matching their body language (they lean forward and you do the same) you will put them at ease and encourage a stronger emotional connection leaving them more open to persuasion. What's in it for them? Show that it is in their best interest if they agree with you. Make them feel that they are onto a good thing where you'll both share in the success.
Tell a story. The tale of the receptionist who spotted the clients watch had stopped, and bought them a replacement battery says more about proactive customer service than twenty bullet points.
Describe the end result. Be sure to show how the new situation will be better than the current one.
Get them to do the work. Ask questions that guide them to your conclusion. People are more likely to agree if they feel that they've come to the conclusion, at least party themselves.
Make your case. State all the facts being clear about the pros and cons. Offer solutions that will resolve their concerns and turn the weaknesses into strengths.
Do them a favour. Tell them what you will do for them in return. They might well feel indebted to you so are more likely to agree.
Use your allies. Refer to the views of someone whose opinion they care about. People are more likely to think or do something when others they respect do too.
Have a plan. Consider the route that you are going to take and what to do with resistance. Be ready to change your plan.