Rock-bottom Anzhi had drawn eight and lost 11 of their opening 19 fixtures, but finally ended their winless run in the top flight thanks to Bukharov's strike seven minutes from time.
Anzhi remain firmly rooted to the foot of the standings but are now just four points behind second-bottom Ural.
Zenit St. Petersburg went to the top of the table but may have handed the advantage to Lokomotiv Moscow following a goalless draw at home to lowly Tomsk.
Zenit and Lokomotiv were level on points before Sunday's match and although Luciano Spalletti's men went a point clear at the top of the standings, Lokomotiv now have a game in hand on their title rivals.
A superb second-half performance helped Dinamo Moscow hit back from a two-goal deficit to storm to a 4-2 win over city rivals CSKA.
Seydou Doumbia's brace in the first half appeared to put the reigning champions on course for all three points, but Kevin Kuranyi scored just after half-time while Christopher Samba levelled matters before the hour mark, with Yury Zhirkov sealing victory for Dinamo with strikes in the 62nd and 69th minutes.