No, Miami was by no means a power in 1968 - certainly not what they were in 1988 - but they WERE ranked in six of the first even AP polls in 1968, getting knocked out after a 31-6 pasting at the hands of (would you believe it?) Auburn.
According to the write-ups I can find from the game, Scott Hunter hit Donnie Sutton with a 74-yard TD pass with 2 seconds left in the first quarter. Mike Dean returned a Pick Six off of Miami QB Lew Pytel.
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Respectfully - you are probably thinking of the 1963 Alabama-Miami game, won by Alabama, 17-12. Jackie Sherrill had a last minute interception of Mira in that game, which was played the same day Notre Dame hired Ara Parseghian as head coach.
That game was played December 14, 1963 - one week after the famous Army-Navy game, and three weeks after the assassination of President Kennedy.
HISTORICAL NOTES
The (then) newly elected President, Richard Nixon, attended the Alabama-Miami football game in 1968.
The day after the 1968 Alabama-Miami game brought one of the most famous games in NFL history: the Heidi Bowl.
The same day as the Heidi Bowl, the Chicago Bears unleashed their backup running back on the Atlanta Falcons. This is because a week earlier, Gale Sayers suffered a season-ending knee injury. He was replaced by a guy named Brian Piccolo, who - less than one year later - would be diagnosed with embryonal cell carcinoma (and die seven months later).