Setbacks against Bologna and title-chasing Inter Milan this month had left Roma 11 points adrift of the top four, with Jose Mourinho facing the first mini-crisis of his reign, having taken charge in the summer.
A backpedalling Abraham headed Jordan Veretout’s corner back towards goal, allowing an unmarked Smalling to nod beyond Spezia goalkeeper Ivan Provedel.
Spezia were undone from another set-piece 11 minutes after the restart when Ibanez headed home from a corner, with the visitors never really threatening an upset as they remain 17th, two points above the drop zone.
Felix Afena-Gyan had a goal chalked off in added-on time and salt was rubbed into the wound as the Roma substitute received a second yellow card and his marching orders for deliberate handball in the build-up, with Mourinho appearing livid on the touchline, but his side moved up to sixth after an otherwise trouble-free win.
There was also late drama in LaLiga at the Estadio Nuevo Mirandilla in Cadiz, who had substitute Salvi sent off late on after being pegged back in a 1-1 draw against Granada.
Santiago Arzamendia had given Cadiz a 32nd-minute lead but Jorge Molina levelled with just two minutes to go before Salvi saw red in the second minute of time added on following a second bookable offence.
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