Shaw's free throw made the score 56-53 and University School missed a long three-point attempt at the buzzer.
Chris Sanchez made two important free throws that gave MLK a late lead.
The fourth-seeded Tigers (14-8) trailed by five points at halftime but used a third quarter comeback to take a 46-38 lead after three quarters.
Peter Hargrove, who led all Tigers scorers with 21 points, scored seven points in the third quarter to lead the second-half comeback.
Shaw had 13 points and nine rebounds, Rajien Griffin scored 11 and handed out eight assists and Dashawn Johnson added seven points and a game-high 19 rebounds for the hometown Tigers.
MLK advanced to Sunday's championship game against either New York Mills (20-3) or Argyle (22-0), the other two semifinal teams.
The University School was the top seed in the tournament with a record of 19-4.
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