Traylor finished with 17 points and 14 rebounds, and Ellis had 14 points and 13 boards for the Jayhawks (25-9), who trailed 56-53 with 9 minutes to go before their game-ending charge.
Glenn Cosey hit five 3-pointers and had 17 points for 15th-seeded Eastern Kentucky (24-10), the Ohio Valley Conference champions. Tarius Johnson and Eric Stutz finished with 15 points apiece.
For most of the game, it appeared as if the Colonels just might follow in the footsteps of Mercer, North Dakota State and the rest of the scrappy underdogs who have crumpled millions of brackets. They led for all but 1:49 of the first half, and turned back the Jayhawks' first big charge of the second before wilting down the stretch.
They simply couldn't contend with the size of Kansas, which had a 43-19 rebounding advantage that translated into a 54-20 edge on points in the paint.
Stanford made the most of its first NCAA appearance since 2008, starting fast and finishing strong to earn a date against Kansas. New Mexico made another hasty exit.
"We came out energetic and ready to go," Randle said after the Cardinal led almost start to finish. "We knew how big this game was. We definitely had enough confidence to finish out the game."
Stanford built an early 16-point lead then held on after New Mexico rallied to tie it midway through the second half. Lemons and Randle each made two free throws in the final half-minute after New Mexico had cut the deficit to two points.
Cameron Bairstow had 24 points and eight rebounds but the Lobos (27-7) got off-days from their other top threats. Kendall Williams and Alex Kirk, who together average 30 points, combined for just six.
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