Alan Ameche, the Baltimore Colt running back who scored in overtime to defeat the New York Giants in the National Football League's 1958 championship game, died of a heart attack on Monday in Houston after undergoing heart bypass surgery last week. He was 55 years old.

Mr. Ameche, who was being treated by Dr. Michael DeBakey, underwent heart bypass and valve replacement surgery on Friday at Methodist Hospital in the Texas Medical Center. He had a similar operation 10 years ago.

His decisive 1-yard plunge into the end zone after 8 minutes 15 seconds of overtime gave the Colts a 23-17 victory over the Giants in what has often been called the greatest football game ever played. A national television audience and a Yankee Stadium crowd of 64,185 witnessed the game on Dec. 28, 1958. It was the first sudden-death overtime game in N.F.L. championship history. 'Most-Remembered Run'

Baltimore began the winning touchdown drive from its 20-yard line, sparked by the passing of Johnny Unitas, the receiving of Raymond Berry and the running of Ameche, who had scored the Colts' first touchdown on a 2-yard run. On the 80-yard drive, Ameche caught an 8-yard pass from Unitas, ran a draw play for 23 yards to the New York 20-yard line and finally slammed over the middle into the Giants' end zone.

''It was probably the shortest run I ever made and the most remembered,'' he said in later years.

Nicknamed The Horse because of the way coaches said he worked in practice, Mr. Ameche, a native of Kenosha, Wis., starred at the University of Wisconsin, where he set a national record by rushing for 3,212 yards and won the Heisman Trophy as the best college football player of 1954. Touchdown at the Start

As Baltimore's first pick in the 1954 draft, Mr. Ameche signed a contract worth $15,000 a year and galloped 79 yards for a touchdown against the Chicago Bears on his first run from scrimmage as a professional.

In his first season, he led the league in rushing, was voted the rookie of the year and was an All-Pro selection. He was All-Pro for three more years and played in five Pro Bowl games before a severe Achilles' tendon injury ended his career after the 1960 season.

In six pro seasons, Mr. Ameche ran for 4,045 yards and scored 40 touchdowns. He was inducted into the National Football Foundation's College Football Hall of Fame in 1975.

After his retirement, Mr. Ameche and Gino Marchetti, a former Colt teammate, founded a successful chain of East Coast restaurants called Gino's. Mr. Ameche later sold out his interest. Mr. Ameche, who lived in Devon, Pa., is survived by his wife, Yvonne, and six children.

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