One thing to remember about "one-vote" victories: they are sometimes not as close as they seem, because sometimes people are reluctant to cast a vote that goes against their party's (or constituents') desires, but will do so *if their vote is needed.*
A good example is the vote on whether to convict and remove Andrew Johnson. Much is made of Edmund Ross' voting to acquit--but if he hadn't, some other Republicans (who voted for conviction in OTL) ptobably would have:
"The closeness of the balloting may in itself be deceiving. Considerable evidence exists that other senators stood ready to vote for acquittal if their votes had been needed. As early as May 18 the Chicago *Tribune* asserted that the President's friends laid claim to four more votes in case of necessity, and the substance of the story was confirmed shortly after the trial by Samuel Randall, the Democratic Congressman from Pennsylvania. On August 3, Johnson himself wrote to Benjamin Truman that [Edwin D.] Morgan [of New York] had been one of the Republicans in question. In 1913, Senator Henderson also asserted that Morgan had been the reputed swing voter. Because of the intense pressure, he voted to convict, but would not have done so had his vote made any difference. Some years earlier the Missouri senator told William A. Dunning that [Waitman T.] Willey [of West Viriginia] had also been ready to switch, a point he later reiterated to Trumbull's biographer, Horace White. He also mentioned Sprague as one of the senators willing to change, and John Bigelow learned that [James W.] Nye [of Nevada] had been another. In short, Johnson's victory was assured long before the vote was taken. A sufficient number of moderate Republicans stood ready to acquit him, come what might." Hans L.Trefousse, *Impeachment of a President: Andrew Johnson, the Blacks, and Reconstruction* (Knoxville: University of Tennessee Press 1975), p. 169.