The 32-year-old striker, who spent the second half of last season on loan at West Ham, steered the ball home from a 90th-minute corner through the legs of Jack Robinson, despite the close attentions of Daniel Agger.
Sunderland transfer target Fabio Borini was named in the Reds' starting line-up alongside Rickie Lambert, but the Italian was forced off with a shoulder injury in the 12th minute.
However, speaking after the match, Reds boss Brendan Rodgers played down the severity of the knock.
"He's only bruised it," said Rodgers. "He won't train now for the remainder of the trip here, but it's only going to be a few days before he's back training and playing again."
In front of a sold-out crowd, Philippe Coutinho was the stand-out player on show for Liverpool, against a Roma team that included former England left-back Ashley Cole.