Estimates of Kolmogorov n-widths d n (B r p , L q ) and linear n-widths n (B r p , L q ), (1 q ∞) of Sobolev's classes B r p , (r > 0, 1 p ∞) on compact two-point homogeneous spaces (CTPHS) are established. For part of (p, q) ∈ [1, ∞]×[1, ∞], sharp orders of d n (B r p , L q ) or n (B r p , L q ) were obtained by Bordin et al. (J. Funct. Anal. 202 (2) (2003) 307). In this paper, we obtain the sharp orders of d n (B r p , L q ) and n (B r p , L q ) for all the remaining (p, q). Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund-type inequalities on CTPHS.