They did. In fact, the idea of allowing emancipated slaves to serve as Union troops was included directly in the text of President Lincoln's Emancipation Proclamation, which reads:

And I further declare and make known, that such persons [emancipated slaves] of suitable condition, will be received into the armed service of the United States to garrison forts, positions, stations, and other places, and to man vessels of all sorts in said service.

The Emancipation Proclamation made it the policy of the U.S. government to enforce the freedom of anyone enslaved within "the States and parts of States" where the population remained in open rebellion against the United States. This meant that Union forces would free enslaved people as they captured territory from Confederate forces. Those slaves who knew about the Union's policy would sometimes escape behind Union lines in order to secure their freedom.

Though there had been some African Americans serving in the Union Army prior to the Emancipation Proclamation, it wasn't until after federal forces began enforcing Lincoln's order that black soldiers, many of them formerly enslaved, began serving to their full capacity. Historian Ira Berlin describes this phenomenon in his book Generations of Captivity:

Like the camps and abandoned plantations, military service offered former slaves an opportunity to realize their own idea of freedom. Although a few irregular black units had been established in South Carolina, Louisiana, and Kansas prior to the Emancipation Proclamation, the vast majority of black men in federal service shouldered shovels, drove teams, and built fortifications, affirming the belief of most white northerners that soldiering was white man's work. When Lincoln's proclamation offered the possibility that black men might be regularly enlisted in the federal army, northern black leaders fanned out over the free states to enlist them, boldly asserting - in Frederick Douglass's words - that once 'the black man gets upon his person the brass letter, U.S.... he has earned the right to citizenship in the United States. [p. 255]

[...]

Over 200,000 black men - most of them former slaves - served as soldiers and sailors in the federal army and navy during the Civil War. That number amounted to about one fifth of the black men of military age in the United States, and a much higher proportion in the North and parts of the South under Union control. [p. 256]