NMAC TURNED INTO A POTENTIAL CONFLICT LTSS BY USE OF TCASII. TCASII SAVES THE DAY AGAIN!

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AT FL350 150 NM S OF YEG HDG 340 DEGS; ATC GAVE US A GARBLED COMMAND. THE FO WAS FLYING; SO I ANSWERED REQUESTING A REPEAT OF THE DIRECTIVE. ATC RESPONDED; BUT WAS STILL UNREADABLE; HOWEVER; I HEARD 'NOW' IN THE COMMAND. IT WAS UNCLR WHETHER ATC WANTED A TURN; A DSCNT; OR BOTH. HEARING THE 'NOW' IN THE VOICE MESSAGE; I WAS IMMEDIATELY ALERTED AND SCANNED THE TCASII. I SAW AN ACFT CONVERGING FROM 90 DEGS TO OUR R AT OUR ALT. I IMMEDIATELY TOOK CTL OF THE ACFT AND DID A DSNDING L TURN TO 320 DEGS. IN THE DSCNT; ATC WAS FINALLY UNDERSTOOD AS WANTING A DSCNT TO FL330. PASSING FL333; I OBSERVED THE OTHER ACFT JUST PASSING OURS; STILL READING FL350 ON THE TCASII. WE NEVER GOT A TCASII ALERT. I FEEL IF WE HAD REMAINED AT FL350; WE WOULD POSSIBLY HAVE COLLIDED. EDMONTON CTR APOLOGIZED; ACKNOWLEDGING THEIR ERROR. I CALLED CTR AFTER LNDG AND WAS INFORMED THE CTLR HAD BEEN RELIEVED PENDING AN INTERNAL INVESTIGATION. I WAS LATER CALLED BY A REPRESENTATIVE OF THE NATIONAL CANADIAN SAFETY BOARD. I GAVE HIM THIS SAME INFO AND DIRECTED HIM TO CALL OUR AIRLINES FOR ANY FURTHER FOLLOW-UP. THERE WAS LITTLE TFC AND IT WAS LATE AT NIGHT. I FEEL THE ATC CTLR WAS PROBABLY A BIT LAX AND SIMPLY DID NOT NOTICE THE TFC CONFLICT UNTIL IT WAS TOO LATE TO PROVIDE NORMAL SEPARATION IN A TIMELY MANNER. WITHOUT TCASII; I DON'T KNOW IF WE WOULD HAVE UNDERSTOOD THE NEED TO DSND IN TIME TO AVOID A POSSIBLE COLLISION; GRANTING THE 2 UNREADABLE VOICE COMS AND THE TIME-CONSUMING NEED TO REQUEST A REPEAT OF THE DIRECTIVE.