After a 1-1 draw after ninety minutes, the Scottish club won the shootout 6-5 when Paul McGowan buried his spot kick following a miss by Sporting's Liedson.
The contest took place on a warm, clear Boston night in front of 32,162 fans dominated by green and white striped Celtic supporters.
After a goalless first half, Celtic's Greek international striker Georgios Samaras opened the scoring in the 70th minute when he converted a penalty past Sporting goalkeeper Tiago after a foul in the box.
The Portuguese side upped the pressure in the final few minutes, peppering Celtic keeper Lucasz Zaluska with a series of chances, before Helder Postiga equalised with a header in the 80th minute.
The exuberant crowd - enjoying football at Fenway for the first time since Brazilian great Pele strode the pitch on July 9, 1968 - mixed a little football tradition with a little baseball, singing both "You'll Never Walk Alone" and "Sweet Caroline," a perennial at baseball's Boston Red Sox games, at half-time.
"It was good to bring the team here today, so the fans can see live what they see on television," said Sporting midfielder Pedro Mendes.
The Red Sox are on a road-trip to the West Coast. Groundskeepers patched over much of the infield to construct the football pitch several yards short of a regulation-sized field.
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