Richie Mo'unga kicked a first half penalty for the only points for the Crusaders, who have claimed seven Super Rugby titles and won all 14 of their games in the competition this season.
Warren Gatland's Lions had been under immense pressure after a mediocre start to their 10-match tour, with an unconvincing 13-7 victory over the Provincial Barbarians and a 22-16 loss to the Auckland Blues on Wednesday.
But a victory born of a huge defensive effort and the almost unerring boot of Farrell will give them a major boost as they head to Dunedin to face the Otago Highlanders on Tuesday.
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(Reporting by Greg Stutchbury in Wellington; Editing by Nick Mulvenney)
