To disagree tactfully, you’d better follow these four steps:

1. Agree to some extent with what the person has said. You may also just apologize.
2. Use a discourse marker like “but”, “yet”, “however”, etc.
3. Use a suitable expression for polite disagreement, e.g. “I don’t see it that way”
4. Give your own opinion.

Expressions for disagreeing politely

• That’s not how I see it. I think … 
• I don’t really agree with you. I believe …
• I’m afraid I can’t agree with you. I suppose … 
• I’m not sure I quite agree with you. To me, …
• Yes, that may be true, but …
• Well, you have a point, but …
• I can see your point, but …
• I see what you mean, but …
• Do you see it that way? I guess …
• I beg to differ.
• Not necessarily. I’d say …
• I don’t see it that way. If you ask me …

Examples

A: I profoundly believe that poverty breeds crime.
B: I’m afraid I can’t agree with you. I suppose poverty alone doesn’t make people commit criminal acts. Lack of education and improper upbringing are the main culprits, if you ask me.
A: Yes, that may be true, but look at residents of deprived areas of the city. As a result of poor living conditions, both parents have to work two or three shifts to make ends meet, and there is nobody to raise the children well or send them to school. Now, don’t you think poverty is to blame?
B: I see what you mean, but still there are many many respectful families in those areas who live a law-abiding life despite financial pressures. If you care enough about your children’s education and devote quality time to their rearing, they won’t grow into criminals later in life.
A: Well, let’s agree to disagree, then.