1973 – The 1st-place Orioles beat the Rangers, 6 – 1, for their 14th consecutive victory. It is the longest winning streak in the American League since 1960, and in the major leagues since 1965. The streak began on August 12th.

This Day In Baseball

On August 27, 1973 — 1973 – The 1st-place Orioles beat the Rangers, 6 – 1, for their 14th consecutive victory. It is the longest winning streak in the American League since 1960, and in the major leagues since 1965. The streak began on August 12th.

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Sources:
Baseball Reference August 27
National Pastime August 27
Retro Sheet
Hall of Fame
SABR Games Project
Replay The Game