I am not a conversation specialist but here are some tips that I have been using and I've found them pretty effective.
1) Be the first one to listen, and be the last one to talk.
Most of the time you can solicit enough information in the discussion and then come up with a summarizing statement. Those kinds of statement are usually harder to debunk because you digested everyone's point of view. In addition, after the summary, which usually can solicit receptive feedback; you can attach a few of your own. In this process, you can also carefully name allies (people whose ideas you want to cite) by quoting what they have said in the setting earlier, making your side more well-supported.
2) Be familiar with common cognitive biases.
Many opinions can be challenged because they are heavily influenced by biases. Here is a list of common cognitive biases for you to read up. When you are listening, look for those kind of anecdotal, unsupported or ill-supported statement, and point out what might invalidate the observation. I wouldn't suggest throwing out the name of biases, the resolution is to gear the person back to rational thinking without them knowing it. Most of the times, I found anecdotal and availability bias the most common problems plaguing people's reasoning. You can easily probe for the reason by asking questions like "interesting point, what made you think that?" and then judge if the reason is sound.
3) Ask the person to elaborate his/her points.
A great, great tool to buy you some time and understand more about their underlying thoughts. More thoughts revealed, more places to find biases or errors.
4) Look for win-win.
Don't look at it as an argument. Treat it as a negotiation, at the end of which the one with more sound reasons and better presentation skills got the lion share. It does not have to all-or-none.
5) Don't start with your sentence with "But..."
Probably one of the most agitating sentences in an argument. No one likes to be blocked sentence by sentence. Acknowledge their thoughts by starting with statement such as: "That seems like a valid point/concern, do we have more data supporting that?", or "That is a great point, besides the color you suggested, I'd also like to tag along with a couple more.", or "I understand why you said that, in the meantime would you be willing to listen to my version?" Then slowly move to the optimal results.
6) Don't shut down an argument/negotiation.
The post-argument regrets that you have been having can easily be avoided if you keep the discussion open. Instead of being speechless, end with something like "That is convincing but I can't tell why I am still on the fence about this. Can I follow up once I have thought it through clearly?" Then really do follow up with your new thought or your acknowledgement of the other person's statements.
7) Know the subject matters.
The basic requirement is that you have to have a good understanding of the issue being discussed. If you have done enough homework, even just bit by bit everyday, the information will be accumulated in your mind along time. Later, your brain can quickly fire up and build some relational arguments, like making a virtual mind map.
8) Develop a habit of critical reading/listening.
Don't just argue/negotiate when you need to. Keep honing the tool by being constantly critical when reading and listening. Develop a habit of thinking in other people's situation.
Generally, I believe an argument doesn't have to be linked to your reflex. This world needs instinctive, fast, extroverted thinkers and it also needs reflective, slow, and introverted thinkers. If reflexive argument is not your strength, come up with a new strategy by slowing down the argument by asking questions, and displaying a sincere attitude to listen/understand. There are arguments that hurt people, and there are arguments that enhance both side's understanding of each other. We should aim for the latter.