Serving for a two-set lead at 5-4, Falla faltered for the first time, but Federer was unable to take advantage, two break points slipping by before the Colombian forced his opponent off court and put away an easy volley on his fourth set point.
However, Bath's advantage had been cut to a meagre twopoints within 11 minutes of the restart as French scrum-half Yachvili coolly slotted over two penalties.
Carter obliged with two more points to give his side a 12-point advantage, however, just when it looked like the All Blacks were running away with it, Australia hit back.
"They obviously have the advantage right now because they've got four points (in their two meetings) and we've got one, so I'm sure they're feeling pretty good, " Chicago's Patrick Kane said.
Parmar saved two set points at 4-5 down in the opening set to force a tie-break and then took advantage of a costly double-fault from his opponent to take it 7-3.