Atalanta moved to 10 points after four games, level with Roma, but the third goal put the Bergamo team top on goals scored.
Koopmeiners was ruthless, particularly from the penalty spot. Deep into first-half added time, Ola Aina took down Brandon Soppy. Koopmeiners converted from the spot. Two minutes into the second half, Torino cleared a free-kick only as far as the Dutch midfielder who smashed the ball home from 25 yards out.
That took Atalanta into a tie for first. Nikola Vlasic knocked the hosts off top spot with a goal on 77 minutes. Koopmeiners had the last word. Valentino Lazaro brought down Atalanta substitute Ademola Lookman and Koopmeiners rolled his penalty right and inside the post.
That brought a joyous end to an evening that got off to an emotional start for the home fans as Josip Ilicic bid them farewell.