When you start a project and get the requirements, there are some things you'd like to change and can't. Sometimes, your manager is one of those things. I don't necessarily think your manager is the only thing that needs attention here. Let's look at what you've provided in some detail.
A recent problem
This is only recent? Did this come out of your manager's year end review for 2013? If it did, take a look at your 2013 projects and see why they didn't get done on time, and learn from those mistakes. Honestly, you should be doing this anyway.
my boss is more worried about how our actions make him appear to the rest of the company than he is about the actual results of our work
Other than aiming for quick wins, do you have other evidence of this? Keeping up appearance for box-checking at review time, and how you appear to the rest of the company are two very different things. For example, my manager takes my team's reputation very seriously, but he doesn't box check. The two do not necessarily go hand in hand.
he is willing to deliver a poor product on time so that he looks good in the eyes of his superiors
Did this happen with just one project? Or with many. If it happened with one, don't think that's a pattern. Just deal with the one project and move on. If it's happening with many, you really need to look at how your estimates are being determined and start adding more time in because if all your team's projects are late, there's a disconnect between the productive capacity of your team and the estimates you're trying to meet. And it's a lot easier to give yourself more time than it is to make your team more productive.
poor reviews for the employees because of all the mistakes
This is very unfortunate. If there's anything that needs an intimate sitdown between the team and your manager, it's this. However, in my experience, most employees don't talk about their reviews, so I have to ask: how many employees are getting bad reviews that you know of (not suspect, but know because they told you or you saw their review for some reason)? Is it just you that you know of?
He seems to be aiming to meet goals that pay off for him in the short run that are causing a loss in the long run
Do you know what his goals are? Has he stated them to you? If so, then there needs to be a discussion (at a level higher than yours) that make sure that his goals are actual representations of what great performance is. That is to say making all your short term goals should put you closer to your long term goals, not further from it. Are these the goals that affect his bonus at the end of the year? If they are, can you blame him? If they're going to give him an extra 4-7% of his salary for meeting these goals, you can damn well bet he's going to do what it takes to meet them. Maybe there needs to be more KPIs, such as support:development ratio to try to bring up code quality, or something like that. But if these are the goals he's been given from upper management, you can't blame him for trying to meet them even if they're the wrong thing to do.
How can I convince my boss that aiming only for wins in the short run will cause him exponentially worse problems in the long run?
You haven't even convinced me that it will. Honestly, this sounds like a situation where you'd rather be doing development work than support work, but you've got a lot of support work to do. All us developers do much more support than we wish we had to. It could be the cost of fixing the bugs first is just not worth it. It might be better for the company to deal with the technical debt than to fix it first. Think of the technical debt like credit cards. You pay down your highest interest cards first right? But you don't know the interest until it gets sent to production. If there's minor bugs in a system that people use once or twice a year, it might not be worth it to fix them -ever- much less before you ship the product.
waiting for his failures to catch up to him
What failures? So far, you've done nothing but describe a manager who is directing his team to meet his stated objectives. I don't know of any managers who set their own objectives, so if there's a failure here, it probably isn't with your manager but those above him.