Goodwin describes this event in Team of Rivals, citing a January 1893 article by Charles Adolphe Pineton, Marquis de Chambrun, a friend of Charles Sumner who accompanied Lincoln to City Point in April 1865 (p. 34):

We were to leave City Point on Saturday, April 8th. A few hours prior to our leaving, the military band came from the headquarters on board the River Queen. We assembled to hear it, After the performance of several pieces, Mr Lincoln thought of the “Marseillaise,” and said to us that he had a great liking for that tune. He ordered it to be played. Delighted with it, he had it played a second time. “You must, however, come over to America,” said he to me, “to hear it.” He then asked me if I had ever heard “Dixie,” the rebel patriotic song, to the sound of which all their attacks had been conducted. As I answered in the negative, he added: “That tune is now Federal property; it belongs to us, and, at any rate, it is good to show the rebels that with us they will be free to hear it again.” He then ordered the somewhat surprised musicians to play it for us.

Goodwin places this event at the window of the White House, however, and includes a citation to Through Five Administrations: Reminiscences of Colonel William H. Crook, Body Guard to President Lincoln by Margarita Spalding Gerry, which gives a remarkably similar quote from Lincoln, at the White House (p. 61-62):

Later in the morning [Monday, April 10] a great crowd came marching into the White House grounds. Every man was cheering and a band was playing military airs. The workmen at the Navy-Yard had started the procession, and by the time it reached up it was over two thousand strong. Of course they called for the President, ans he stepped to the window to see his guests. When the cheering had subsided he spoke to them very kindly and good-naturedly, begging that they woud not ask his for a serious speech.

“I am going to make a formal address this evening,” he said, “and if I dribble it out to you now, my speech tonight will be spoiled.” Then, with his humorous smile, he spoke to the band:

“I think it would be a good plan for you to play Dixie. I always thought it was the most beautiful of our songs. I have submitted the question to our Attorney-General, and he has given his legal opinion that we have fairly earned the right to have it back.” As the opening bars of Dixie burst out, Mr. Lincoln disappeared from the window. The crowd went off in good humor, marching to the infectious rhythm of the hard-won tune.

Should be noted that Crook was on the City Point trip, as well.

Finally, it is described in Collected Works of Abraham Lincoln, as being written up in the National Intellegencer (p. 393):

“The procession proceeded along Pennsylvania avenue gaining accessions at every step, despite the mud and rain, and when it turned up Fifteenth street it is estimated that there were over three thousand persons in the crowd. The procession proper—that is, those who had come from the Navy Yard—and a portion of the crowd proceeded to the residence of Secretary Welles, while the other portion kept along Pennsylvania avenue to the White House and the War Department. At the latter place the band of the Quartermaster’s regiment, Capt. Tompkins, under the leadership of Prof. Blish, and the band of the Fourteenth regiment V.R.C., were stationed, and their excellent music attracted an immense concourse of people, who called again loudly for Secretary Stanton, but failing to get him out, the crowd, preceded by the Quartermaster’s band, moved toward the White House, and in a few moments an immense number of people were assembled, and completely filled the portico, the carriageway, and pavements on either side, while many were forced to content themselves with a stand-up place in the mud. The bands played, the howitzers belched forth their thunder, and the people cheered. Call after call was made for the President, and his failure to appear only made the people cry out the louder. Master Tad Lincoln, who was at the window, appeared to hugely enjoy the shouting, cheering, and swaying to and fro of the crowd, who evinced a determination not to depart until the Chief Magistrate acknowledged their greeting by his presence. At length, after persistent effort, the presence of Mr. Lincoln was secured. Three loud and hearty cheers were given, after which the President said:

“FELLOW CITIZENS: I am very greatly rejoiced to find that an occasion has occurred so pleasurable that the people cannot restrain themselves. [Cheers.] I suppose that arrangements are being made for some sort of a formal demonstration, this, or perhaps, to-morrow night. [Cries of `We can’t wait,’ `We want it now,’ &c.] If there should be such a demonstration, I, of course, will be called upon to respond, and I shall have nothing to say if you dribble it all out of me before. [Laughter and applause.] I see you have a band of music with you. [Vocies, `We have two or three.’] I propose closing up this interview by the band performing a particular tune which I will name. Before this is done, however, I wish to mention one or two little circumstances connected with it. I have always thought `Dixie’ one of the best tunes I have ever heard. Our adversaries over the way attempted to appropriate it, but I insisted yesterday that we fairly captured it. [Applause.] I presented the question to the Attorney General, and he gave it as his legal opinion that it is our lawful prize. [Laughter and applause.] I now request the band to favor me with its performance.”

“In accordance with the request, the band struck up `Dixie,’ and at its conclusion

So. The local paper confirms the second account, the event taking place at the White House. The two eyewitness memoirs, both written long after the event, describing very similar occurrences and quotes, happening 48 hours apart. Both apparently credible witnesses. I suspect that Goodwin, having contemporaneous confirmation of the White House event, decided that Chambrun was mistaken about the date. Perhaps, but given the tendency for modern day pols to tell the same joke or anecdote over and over again at public appearances, it wouldn’t surprise me too much if Lincoln actually did this twice.