Saracens exploded at the start and scored two tries in 14 minutes, but Connacht battled back to square things up at 17-17 by the break. Then it took Farrell's unflinching nerve, and deadly accurate boot, to seal four priceless away points in Pool 3.
"It is always difficult to come away in the Heineken Cup and we had to dig deep and fight for what we got. They seemed to grow another leg when they came back into the game and the crowd got behind them," he said.
"It is a difficult place to come and play and there was a big wind out there in the first half. It was my first time in Galway, but it was a brilliant experience.
"We had quite a few opportunities, and could have scored three tries in the early stages of the second half, but a mistake or two, or a penalty against us, forced us back into our own half. We will learn from our mistakes today.
"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. Maybe we got it a bit wrong in the first half and we got pinned back.
"When we had good field position and played in the right parts we had a number of chances. I think we will learn from our mistakes and we will examine them, prepare as ell s we can in the week and be ready for Toulouse next weekend."
Saracens play four time winners Toulouse at Wembley in round two next weekend. The French giants scored six tries in a 38-5 home win over Italian franchise, Zebre.