De Rossi made a dream start to life at Boca after leaving boyhood club Roma, marking his bow with the 28th-minute opener on Tuesday.
The 36-year-old's bullet header at Estadio Ciudad de La Plata had three-time Copa champions Boca on track to advance to the round of 16.
However, when De Rossi was substituted with 13 minutes remaining, Boca lost control against the second-tier side.
Juan Martinez equalised for Almagro within five minutes of former Italy international De Rossi's exit as the match went to extra time.
De Rossi then watched from the sidelines as Boca – winners in 2014-15 – were stunned 3-1 in the shoot-out.
Afterwards, Boca forward Ramon Abila told reporters: "It bothers me to lose, but we apologise to the people of Boca."