Brendan Rodgers and Rudi Garcia both named strong sides for the clash at Fenway Park, and it was the Italians who earned a late victory through the Dane's unfortunate strike
Both managers named strong sides for the clash in Boston, with summer signings Rickie Lambert and Emre Can both featuring for Liverpool and Ashley Cole in the starting XI for Roma.
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However, it is an injury concern that will worry Brendan Rodgers most, with Fabio Borini having to be substituted in the first half following a blow to his shoulder.
Linked with a move to Sunderland, Borini went down under a challenge and landed awkwardly before being substituted in the 12th minute. The Italian was expected to make a £14million move to Sunderland, where he enjoyed a successful campaign on loan last season, but that move could now be in doubt.
One of Rodgers' four signings of the summer so far, Can, replaced Borini just days after suffering a calf injury in Liverpool's win over Preston North End.
But it was Roma's marquee signing, ex-Chelsea man Cole, who created the first chance of a slow first half in the 33rd minute.
The former England left-back crossed from the left for veteran Francesco Totti, who volleyed just wide of Jones' goal from close range.
There was more life to the contest after the break and another Liverpool signing, Rickie Lambert, went close to opening the scoring, as he tested Lukasz Skorupski from just inside the area, with the Polish goalkeeper just up to the task.
Multiple substitutions then 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 soon after, however, as Agger's touch took a well-worked corner past Jones and into his own goal to earn the Serie A side a late - if slightly fortunate - victory.