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Let <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be a Euclidean sphere of curvature <inline-formula><math display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula> be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><msub><mo>×</mo><mi>f</mi></msub><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> is a smooth positive function on an open interval <i>I</i> of <inline-formula><math display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> to Riemannian warped products <inline-formula><math display="inline"><semantics><mrow><mi>I</mi><msub><mo>×</mo><mi>f</mi></msub><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.