Burnett had played in only three games previously, but he became the fourth player to lead the young Hurricanes in scoring this season. The 6-foot-2 guard added eight rebounds in 24 minutes.
Miami, which has never previously been ranked so high so early in a season, improved to 8-0. The Illini fell to 6-1 with their earliest loss in the past four years.
Illinois came into the game averaging 90 points, but shot only 33 percent, including 7 for 30 from 3-point range. Rayvonte Rice led the Illini with 22 points and 12 rebounds, and Malcolm Hill added 13 points, but the rest of the team shot 23 percent.
The Hurricanes won despite shooting only 38 percent, including 9 for 30 from 3-point range. But they went 17 for 21 at the free-throw line, while Illinois went 8 for 12.
Miami was up 37-24 at halftime for its largest lead, but Illinois scored the first 10 points of the second half. The Illini cut the deficit to 48-46 but could get no closer.
Three-pointers by Omar Sherman -- his only basket -- and Rodriguez helped the Hurricanes regain their cushion. They sealed the win by sinking five of six free throws in the final minutes, and in the closing seconds the crowd of 6,086 chanted, "A-C-C! A-C-C!"
Illinois shot only 25 percent in the first half, including 3 for 17 from 3-point range.
Illinois: Rice continued his streak of scoring at least 14 points in every game this season.
Miami: The Hurricanes went 22 for 58 after shooting better than 50 percent in each of the past four games.
Illinois hosts American on Saturday.
Miami hosts Wisconsin-Green Bay on Saturday.