Ibehre looked to have won it for the Cumbrians in normal time but Mani Dieseruvwe levelled late on before being sent off. Ibehre and Osei struck in the second period of extra-time.
Carlisle’s goal came under an early siege as Dan Gardner rattled the woodwork with a low effort.
But as Carlisle settled it took a fine save by Spireites goalkeeper Tommy Lee to keep out a curling effort from Angelo Balanta to keep his side on level terms in the 37th minute.
Twice more before the break Lee stood between Carlisle and the lead as he dealt with Danny Grainger free-kicks from outside the area.
However, he could do nothing with the header by Ibehre which gave Carlisle a 75th-minute lead. The cross came from fellow sub Osei on the left enabling Ibehre to nod the ball downwards inside the goalkeeper’s left-hand post from six yards.
Chesterfield sub Dieseruvwe lashed a shot past Mark Gillespie from 15 yards to level for the Spireites. But it was a mixed night for Dieseruvwe who saw red in the second minute of extra-time for a foul on Michael Raynes.
Ibehre struck for the second time in the 107th minute, hammering a tremendous volley past the stranded Lee after a long ball in from the left and Osei clinched it with a solo goal right at the death.