But the Lobos hit their final eight foul shots to hold off the upstart Flames (1-1).
Hugh Greenwood had nine points and nine rebounds for New Mexico, which shot just 27.8 percent (5 for 18) in the second half.
Hayden Humes led UIC with 16 points, Gary Talton added 14 and Barnes 11.
New Mexico will face George Mason on Sunday, while UIC faces Mercer on Saturday.
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- New Mexico