Seton Hall had lost its last two games after a nine-game winning streak that covered the whole month of February.
"We came out loose and I think I got everybody a lot more comfortable and that showed with well-balanced scoring," he said.
The Pirates (17-11), who many people felt needed one win in this tournament to ensure an at-large berth in the NCAA field of 65, will play Connecticut (19-8) in the quarterfinals on Thursday night.
"I hope this did it but I'm not on the committee," Barrett said. "But we have more games to play here and that's what we're in this for right now."
Kelly Whitney had 17 points on 7-for-9 shooting and Seton Hall shot 62 percent in the first half to take command.
"This was a game where we had a chance to get out and run," Pirates coach Louis Orr said. "We hadn't done that the last couple of games."
Andre Sweet had 16 points for Seton Hall, while John Allen had 15.
Paulo Coelho had 12 points for the Hurricanes (11-17), who finished the season losing seven of their last eight games.
In addition to the Pirates' torrid first-half shooting (15-for-24), they held Miami to 21.7 percent (5-for-23) to take a 38-19 lead.
"Holding them to 19 points, when we play defense like that usually good things will happen," Orr said.
The lead reached 55-34 on a free throw by Allen with 7:34 left, and Miami didn't get closer than 15 points the rest of the way.
The Pirates finished at 48 percent for the game (25-for-52), while the Hurricanes shot 33 percent (17-for-51).
"It was disappointing for us from the standpoint that we had early opportunities and didn't take advantage of any of them," Miami coach Perry Clark said. "When we didn't do that they started doing the things that led them to so much success this year and it snowballed on us."
Seton Hall lost to St. John's in the opening round of last year's tournament, while Miami reached the semifinals, losing to Pittsburgh.
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