Jose Mourinho’s side took advantage of the opportunity to fight for a top-four position after Juventus were deducted 15 points for the current season, leaving the former third-placed club marooned in mid-table.
Paulo Dybala ran into the box in first half stoppage time and picked out El Shaarawy with a cross who finished past Spezia goalkeeper Bartlomiej Dragowski.
Abraham doubled their lead four minutes after the break, getting through the Spezia defence and calmly slotting the ball into the lower left corner of the goal from the edge of the box.
The result left Roma level on 37 points with third-placed Inter Milan, who play Empoli at home on Monday. Spezia are in 15th place.
Juventus play Atalanta later on Sunday, while Lazio and Milan face each other on Tuesday.
Meanwhile Napoli further extended their grip on the Serie A title race on Saturday by beating local rivals Salernitana 2-0 to temporarily move 12 points ahead of the chasing pack.
