GE had an engine (GE90-115) that was already developed and certified by EASA & FAA. Rolls Royce did not. That meant GE could and did offer a better deal to Boeing looking at both timing and total aircraft certification cost which permitted Boeing to get this plane into revenue service sooner at lower cost than other alternatives.
This is true, but RR were willing to go into the 777X program without exclusivity, unlike the other two. GE offered financial incentives to be a risk-sharing partner and that is what went ahead, in hindsight, an extremely good result for GE.
When the final A-350 design was released, the proposed GEnx engine for an earlier A-350 design concept was withdrawn as the final A-350 aircraft competes against the Boeing 777 ER/LR (GE90-115). It is business strategy at play, sometimes you win, sometimes you lose.
Just to add to that, GE were willing to power the smaller 2 A350s in its current incarnation, but Airbus wanted all 3, or nothing.