ACR MLG FLEW PAST THE ARPT ON A VISUAL APCH AT NIGHT.

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AT 10 MILES NORTH OF MSP; I ACCEPTED CLRNC FOR A VISUAL APCH TO RWY 11L WHILE DESCENDING FROM 6000' WITH THE ARPT TERMINAL BLDG AND ENVIRONMENT IN SIGHT; BUT NOT THE RWY ITSELF. THE F/O; FLYING THE ACFT; MANEUVERED TO A LEFT BASE LEG AT 3000'; HEADING ABOUT 200 DEGREES. WE LOWERED THE LNDG GEAR AND ACCOMPLISHED THE FINAL LNDG CHECK LIST. I THEN LOOKED FOR THE ARPT AGAIN AND OBSERVED A ROTATING BEACON AT OUR 10 O'CLOCK POSITION. I ASKED THE APCH CTLR THE LOCATION OF THE ARPT AND HE CONFIRMED THAT IT WAS AT 10 O'CLOCK; SIX MILES. WE CONTINUED TOWARD THE ROTATING BEACON WHICH; IN FACT; WAS AT THE FLYING CLOUD; MN ARPT; AND WERE HANDED OFF TO THE LOCAL (TWR) CTLR. I TOLD HIM WE DID NOT HAVE THE RWY IN SIGHT; AND HE STATED THAT HE WOULD TURN UP THE INTENSITY ON THE RWY LIGHTS; AND CLRED US TO LAND ON RWY 11L. AT THAT POINT; I NOTICED THAT MY ADF POINTER; IN THE VOR POSITION; WAS POINTING TO OUR 8 O'CLOCK POSITION; AND THAT THE 11L BACK COURSE ILS; ON THE FIRST OFFICER'S INSTRUMENT; WAS BEHIND US. REALIZING THAT WE HAD PASSED THE MSP ARPT; I SO INFORMED THE CTLR AND REQUESTED A VECTOR BACK TO THE TFC PATTERN. HE VECTORED US LEFT TO 060 DEGREES; WHICH WE DID. I TOLD HIM WE STILL DIDN'T SEE THE RWY; AND HE SAID HE WOULD TURN THE RWY LIGHTS UP MORE. WHEN HE DID THAT; WE IMMEDIATELY SAW THE RWY; ENTERED A RIGHT BASE LEG; AND LANDED. AFTER LNDG; THE CTLR REQUESTED THAT I CONTACT HIM BY PHONE; WHICH I DID. DURING OUR CONVERSATION; HE REMINDED ME THAT THE MSP ARPT WAS NOT EQUIPPED WITH A ROTATING BEACON. I SUGGESTED THAT A VECTOR BACK TO THE ARPT AT THAT POINT WAS PREFERABLE TO LNDG AT THE WRONG ARPT. THE AREA SUPERVISOR ENTERED THE CONVERSATION AND INFORMED ME THAT BECAUSE WE HAD PASSED THE RWY; THAT THIS WAS A REPORTABLE INCIDENT. OUR ACTIONS NOTWITHSTANDING; I BELIEVE THAT THE LACK OF A ROTATING BEACON AT MSP; AND THE RWY LIGHTS BEING IN A VERY DIM INTENSITY POSITION DURING VISUAL APCHS WERE CONTRIBUTING FACTORS IN THIS INCIDENT. SUPPLEMENTAL INFORMATION FROM ACN #86282: I COULD NOT SEE THE RWY LIGHTS WITH ALL THE OTHER CITY LIGHTS; AND TOLD THE CAPT SO. LOOKING AT MY NAVIGATIONAL AIDS; I CONFIRMED WE WERE NW OF MSP AND I STARTED A TURN TOWARD FINAL. THE CAPT TOLD TWR WE HAD LOST THE ARPT AND THE TWR SAID THEY WOULD TURN UP THE LIGHTS. THE CAPT ASKED IF THE ARPT WAS AT 10 O'CLOCK. TWR; HESITANTLY SAID; 'YES'; AND I LOOKED AT MY INSTRUMENT AND CONFIRMED IT WAS NOT MSP. I ASKED FOR A HEADING AND THEY TURNED US TO 060 DEG. THE RWY LIGHTS WERE TURNED BACK DOWN BY THIS TIME; BUT WE SAW THE RWY AND LANDED UNEVENTFULLY.