The Cardinal (41-23-2) advanced to the Bracket 1 Finals against Georgia, needing to beat the Bulldogs twice to return to the best-of-three championship round for the second time since 2003.
Starter Danny Sandbrink and reliever Erik Davis held Miami's the potent offense in check. The Hurricanes (53-11) failed to join the 1999 squad as the only No. 1 national seeds to win the College World Series.
Miami, ranked No. 1 for most of the second half of the season, has been eliminated in three games in each of its four College World Series appearances since 2003.
Phelps collected three of Stanford's 11 hits -- a single, double and triple -- and drove in two runs.
After Miami's first two batters reached base, Sandbrink and Davis combined to hold Miami without a hit until the sixth.
Davis (8-3), sent to the bullpen after struggling as a starter late in the year, shut down a Hurricanes threat when he entered in the fifth. After Sandbrink issued a leadoff walk, Davis walked his first batter to put two runners on.
But he got Jemile Weeks to foul out trying to bunt and struck out Miami's top sluggers, Yonder Alonso and Mark Sobolewski.
For Alonso, Cincinnati's seventh pick in the draft, it marked the first time since March 2007 that he struck out three times in a game.