Linked with a move to Sunderland, Borini went down under a challenge and landed awkwardly on his right shoulder before being substituted in the 12th minute at Fenway Park.
Borini, who has been unable to cement a place at Anfield since joining in July 2012, came off holding his shoulder.
The Italian was expected to make a £14 million move to Sunderland, where he enjoyed a successful campaign on loan last season, but that move could now be in doubt.
Marco Borriello claimed what proved to be the match-winner, side-footing in a corner in the 90th minute.
One of Rodgers' four signings so far, Emre Can, replaced Borini just days after suffering a calf injury in Liverpool's win over Preston North End.
New signing Ashley Cole started for Roma and created the first chance of a dull first half in the 33rd minute.
The former Chelsea left-back crossed from the left for Francesco Totti, who volleyed a difficult chance wide of Brad Jones' goal from close range.
It was a rather forgettable first half as two well-organised defences meant both teams struggled to create opportunities.
There was more life to the contest after the break and another Liverpool signing, Rickie Lambert, went close to opening the scoring.
Lambert looked the most likely and tested Lukasz Skorupski from just inside the area, but the Polish goalkeeper was up to the task.
Both sides failed to construct good chances as they often broke down approaching the attacking third.
Multiple substitutions disrupted the flow of the match, but Roma forward Antonio Sanabria almost struck when he forced Jones into a save down low to his right in the 89th minute.
Rudi Garcia's men would get the goal they wanted as Borriello scrambled in the winner, which went through Jack Robinson's legs near the goal-line, from a corner.
