It depends quite a lot on jurisdiction and situation, but as a general thing that wouldn't work. The Fugitive Slave Clause and the Fugitive Slave Act of 1850 remained the law of the land upon the outbreak of the Civil War. Just setting foot on free soil doesn't emancipate anyone, but some states do have laws that grant freedom to slaves who are brought willingly by their enslavers into free jurisdictions. If the slaves know about those laws, or local antislavery whites and/or free blacks (more often the latter) catch wind of their presence and make them aware, then they might be able to get a freedom suit going based on them but it's not by any means an instant thing.

Fugitive renditions, covering those enslaved people who stole their bodies, continue and probably increase somewhat in the immediate opening of the war. As part of the quest to defuse the secession crisis and keep the Upper South loyal, Lincoln pledged to strictly enforce the Fugitive Slave Act even though it's one of the most hated laws around in most of the North. The idea was to show the Upper South that the Republicans aren't out to get them and respect their right to keep on with slavery within the Union, while maybe convincing some of the Cotton states that this is all a big misunderstanding.

For some time, even military lines are fairly porous when it comes to rendering up slaves who ran to them. At first it appears that US officers returned fugitives almost as a matter of course, but as it became clear that this war would go on for more than a weekend or two and that the rebels would use enslaved labor for military purposes it looks increasingly unwise -as well as sometimes morally repugnant, a lot of northerners get their first up-close look at slavery here- to turn those people back over to their enslavers.

Benjamin Butler, an otherwise undistinguished and infamous general, reasoned that fugitive slaves who came to Union lines presented a novel legal situation. If the rebel states were outside the Union, as they said, then the Fugitive Slave Clause and Act did not apply to them anymore. Those are creatures of American law and the rebels are foreigners. They were also levying war against the United States and using slaves to that end. Thus, if slaves were property then they could be seized by a belligerent power as contraband of war. So he did that. Within a relatively short period, Butler's policy becomes Army policy.

But getting to US lines does not necessarily make you free. The proper status of the contrabands is ambiguous. Legislation authorizes confiscating the slaves of disloyal owners, but there's some doubt as to whether it would withstand a peacetime court challenge. The Emancipation Proclamation frees slaves who cross US lines effective January 1, 1863, but it's an assertion of presidential war powers which also may not survive the courts.

None of that stuff applies to the slaves of people who live within states which remain loyal. If you're in Kentucky or Maryland, you're flat out of luck. ...unless you lie about it. It turns out that military officers in the middle of waging a war have a whole lot better things to do than sort out if a person claiming to be property of a traitor actually is or not. Those things include seeing if the men among them would be interested in working for the US military, either in a support role or as an actual soldier. With armies on the move all the time, a lot of opportunities open up for enslaved people to find some freedom.

All that said, you have to be willing to take some risks and have the military near enough at hand. As a result, there are still slaves in Kentucky and Delaware until the ratification of the Thirteenth Amendment makes everything done during the war vis-a-vis ending slavery legal, whether it was beforehand or not. Ratification was proclaimed on December 18, 1865. For context, the big Confederate surrenders took place back in April.

Sources

All of this is discussed in some detail in Eric Foner's The Fiery Trial. Prewar fugitive slave renditions are also a subject of his Gateway to Freedom, about the Underground Railroad. Don Fehrenbacher's The Slaveholding Republic also treats the Fugitive issue extensively, from the Constitution onward.