A B737 FLC HAS A NEAR RWY INCURSION WHEN THEY PASS UP THE TXWY TURNOFF IN A NIGHT OP AT CLE.

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NORMAL PUSH FROM CLE. CLRED TO TAXI VIA TXWYS S AND L TO RWY 6L; HOLD SHORT OF RWY 6R. PUSH WAS LATE DUE TO PAX THAT ELECTED TO GET OFF AT THE LAST MIN. (HAD TO REMOVE BAGS.) TAXIING AT NIGHT ON TXWY S. DID NOT SEE THE TURN ON SIGNS TO TXWY L. VERY ASTUTE GND CTLR DIRECTED A TURN; WHICH PREVENTED US FROM ENTERING RWY 6R. COMPLETED 180 DEG TURN AND ENTERED TXWY L. NORMAL TAXI AND TKOF. BOTH PLTS WERE HEADS UP DURING TAXI. BOTH WERE LOOKING FOR TAXI SIGNS. TAXI WAS SLOW WHICH ALLOWED US TO STOP AND AVOID A RWY INCURSION. ONLY OTHER CONTRIBUTING FACTOR MAY BE THE WIDTH (VERY WIDE) ENTRY ON TXWY S. WE SHOULD HAVE SEEN THE RWY SIGNS; BUT THAT ISN'T WHAT WE WERE LOOKING FOR. NEXT TIME; DEFINITELY NEED TO IDENT RWYS AND WORK BACK TO TXWYS. I HAD ALREADY DELAYED CHKLIST READING SO THAT WAS NOT A FACTOR. NOTE: 1) DON'T KNOW IF THIS IS A 'TROUBLE' AREA FOR RWY INCURSIONS AT NIGHT; BUT THE CTLR WAS OBVIOUSLY WATCHING US AS WE TAXIED. TO ME; THE AREA IS VERY SIMILAR TO THE AREA IN BWI THAT NOW HAS LIGHTS ACROSS THE HOLD LINE. 2) WE DID NOT CROSS THE HOLD SHORT LINE. 3) THERE WERE NO ACFT CONFLICTS. SUPPLEMENTAL INFO FROM ACN 548724: GND CTL DIRECTED US TO 'HOLD POS' AS WE WERE APCHING THE DEP END OF RWY 6R. ACCOMPLISHED A 180 DEG TURN AND REMAINDER OF THE FLT WAS UNEVENTFUL. I ESTIMATE WE WERE ABOUT 100 FT FROM A RWY INCURSION. I LIKE TO BELIEVE THAT THE CAPT OR MYSELF WOULD HAVE CORRECTED OUR MISTAKE EVEN WITHOUT GND CTL INTERVENTION; BUT IT WAS DARK AND THE RWY MARKER WAS JUST COMING INTO VIEW. I DON'T FEEL WE WERE DISTR BY CHKLISTS OR ACCOMPLISHING PERFORMANCE DATA. IT IS A VERY QUICK L TURN ONTO TXWY L AND WE SIMPLY MISSED IT. AS THEY TAUGHT US IN TRAINING; KEEP YOUR HEAD OUT OF THE COCKPIT WHEN TAXIING AND PAY ATTN!