Gibbs made 13 of 18 shots, including six 3-pointers in eight attempts, none bigger than the one with 36 seconds left after the Eagles pulled within 77-74 on Anthony Elechi's free throw 22 seconds earlier.
Miguel Dicent hit a 3 for Morehead State with eight seconds left but Gibbs, who went 9 of 12 from the line, clinched it with a free throw at two seconds.
The only Wildcat to score 40 points other than Stephen Curry, who did it six times, Gibbs hit the 35-point plateau for the fifth time, four coming this season.
Jordan Barham added 17 points for the Wildcats (8-2), who shot 60 percent from the field and made 10 of 21 3s.
All five starters scored in double figures for the Eagles (6-4), topped by Miguel Dicent with 16 and Brent Arrington with 15. Morehead State made 11 of 23 3s.