It appears that this quotation is probably apocryphal.

Du Bois himself does no more than hint at a source – he prefaces the quote with the remark that "One has but to read the debates in Congress and state papers from Abraham Lincoln down to know that the decisive action which ended the Civil War was the emancipation and arming of the black slave..." The sentence does not appear in the eight volumes of Clifford and Miller's The Works of Abraham Lincoln, nor in Don and Virginia Ferenbacher's The Recollected Words of Abraham Lincoln, which are the two main sources for verified Lincoln quotations. A search of the Google Books archive for the 19th century reveals no published mentions of the quote before Du Bois used it; and no 20th century work that quotes the phrase that I can find contains a source citation for it other than referencing either DuBois, or some later writer.

Furthermore, those who do use the quote rarely date it precisely. The closest we get is Philip S. Foner's History of Black Americans, vol.3 p.455, which suggests the quote was made in a "victory speech" that Lincoln delivered after Appomattox, in April 1865. Jabari Gravy, in his The Fallacy in the Promise, and the National Park Service page devoted to Lincoln's proclamation to establish a "Bureau of Colored Troops" both merely give the date 1865.

Foner's reference suggests the quote ought to appear in the address Lincoln delivered on 11 April, two days after Lee's surrender; in fact, given that the sentence is in the past tense, it can only have been said between Lee's surrender on 9 April and Lincoln's death on 14 April. The address of 11 April was actually the president's last speech, one that discussed his plan for reconstruction, and it is the subject of an entire book – Louis P. Masur's Lincoln’s Last Speech: Wartime Reconstruction and the Crisis of Reunion. But it does not appear in Masur's work, nor is it in the full text of the speech, archived at Abraham Lincoln Online – which, by the way, shows that Lincoln used the terms "colored men" and "freed-people", not "black freedmen" – on this occasion, in any case.

I can't be 100% certain about this conclusion, since even sources explicitly devoted to examining fake Lincoln quotes (such as Edward Steers's Lincoln Legends: Myths, Hoaxes and Confabulations Associated With Our Greatest President and Paul F. Boller's Quotemanship: the Use and Abuse of Quotations for Polemical and Other Purposes) or to contextualising Du Bois's writings (such as Nick Bromell's A Political Companion to W. E. B. Du Bois) do not tackle the problem of where he got the quotation from. Boller's book sources the quote to "Mrs. Brodie" (Fawn M. Brodie), who of course wrote well after Du Bois, and not even explicitly about Lincoln – she was actually writing on Jefferson and his slave-holding.

Someone with a more detailed knowledge of Lincoln's policies and words than I have may perhaps be able to suggest a source from which the quote may have been paraphrased.