The fly-half, who scored 51 points on the summer tour to Australia, praised the Connacht fight throughout the match.
"It is always difficult to come away in the Heineken Cup and we had to dig deep and fight for what we got," he told ERC Rugby.
"They seemed to grow another leg when they came back into the game and the crowd got behind them.
"The effort and fight we showed was brilliant, especially when we were defending on our own line, and we will be a better team for this experience."
Farrell wasn't the only member of the 2013 Lions squad to get on the scoresheet, as flanker Sean O'Brien and winger Tommy Bowe crossed the try-line in Leinster and Ulster's respective wins.
Lions captain Sam Warburton saw his Cardiff Blues side outplayed by Exeter Chiefs, as Leigh Halfpenny scored nine points in the 44-29 defeat.
England prop Alex Corbisiero and Wales wing George North could not inspire Northampton to victory, as they fell 19-13 to Castres.
And Irish winger Simon Zebo hobbled off with an ankle injury as Munster slumped to defeat against Edinburgh.