The use of the closer on the road, when extra innings are looming, is actually one of the manager tactical nuances that most fascinates me. Conventional wisdom told "old school" managers always to hold back the closer in case the team took a lead in extra innings. This helped closers to get a lot of rest as they sat idly by and watched inferior relievers give up walk-off hits like they were going out of style.
However, the opposite mind-set -- always put your closer in for the bottom of the 9th -- is also flawed. I believe there are a set of variables (discussed below) that should be considered as to when you put your closer in to try to preserve a tie. Remember that in order to win a game on the road that is tied after 8.5 innings, either your closer will have to go more than 1 inning or, more likely, another reliever or two are going to have to step up.
Note that the default position should, in fact, be to use your closer to try to get a tie game into extra innings. Every new inning you force is a chance to put up a crooked number that most any reliever can preserve. And better to ask a mediocre reliever to try to save a one-run lead in extra innings than to watch him produce a one-run deficit in the bottom of the 9th. In general you want to use your best reliever, then your next best reliever, and so on, in a game that is clearly winnable and is even more clearly quite loseable any minute.
Here are the factors I would be considering were I the manager whose team was on the road, tied going to the bottom of the 9th:
Has the other team already used its closer?
Usually the answer is "yes" because most teams will use their closer in the top of the 9th of a tie game knowing that there are no potential save situations for the home team in extra innings. If the home team uses its closer, you are always going to be on at least even footing if you use your closer in the bottom half -- your closer may not be available for the 10th but neither will be the other team's closer. If the other team wants to use its next best available reliever for the 10th, you have yours ready for the bottom half. The other team has to play its hand first and you are playing at no disadvantage as you will always have the equivalent (or better, if the other team does not use its relievers from "best to worst") to play in return.
Has your closer been heavily worked lately?
One reason not to go with your closer unless you take an extra innings lead is if your closer is borderline on rest. If your closer has worked two days in a row, or three of the last four, or threw 25 pitches the night before, perhaps he is available but all things being equal you would prefer to stay away from him.
In this scenario you might hold your closer back, using him only if you do make it to extra innings and take a lead, but allowing the rest of your bullpen to take its best shot at keeping you even and only taxing your closer if a save situation presents itself.
This avoids the lose-lose scenario where you use your closer even though he could use a day off, only to watch another reliever lose the game later -- either by giving up the game's next run or by blowing a lead gained after your closer has come and gone.
What part of the order is due up?
Great relievers can plow through any part of a batting order, but for most relievers you would be a lot more comfortable watching them try to navigate the 7-8-9 hitters than the 3-4-5.
If the bottom of the order is due up in the bottom of the 9th, you might let a "lesser" reliever take the 9th, holding your closer back for the 10th when the top or heart of the order will bat. If your non-closer can handle the 9th, this allows you to "buy" an inning for your team to try to score in the top of the 10th and create a save situation, or for your closer to buy you an inning by getting you to the 11th.
Similarly, if the bottom of the 9th is going to feature a series of LH batters you might choose the 9th to insert your LOOGY instead of your "best reliever," gauging that this is the more opportune time to play that card -- and with it a chance to hold your closer back an inning.
Remember that if the other team uses its closer in the top of the 9th and you can hold your closer back until the bottom of the 10th, you have a substantial advantage in the matchups each inning. In other for this to work, though, you have to get through that bottom of the 9th, so you had better have good reason to think someone other than your closer is well equipped to get through it.
When you run through this checklist, the lion's share of the time you are going to conclude that the best time to put in your best reliever is in the bottom of the 9th. Usually, but not always. As the season progresses, see how often Bob Melvin's choices reflect attention to these factors, as they are the ones I believe provide the best balance between the flawed 9th inning paradigms of "never' and "always".