Assuming that the given transfer function describes a causal system, it is true that the given system is stable because all its poles lie in the left half-plane. This type of stability referred to here is BIBO stability (bounded-input, bounded-output stability). BIBO-stability is also called external stability because it only describes the relationship between signals observable at the inputs and outputs of the system. This is different from a system's internal (asymptotic) stability. A system can be BIBO-stable but internally (asymptotically) unstable. BIBO-stability says nothing about the internal behavior of a system. A complete system characterization is given by the state-variable description, which is an internal description of the system, revealing whether a system is asymptotically stable.
An example of a BIBO-stable but asymptotically unstable system is the following causal system consisting of two concatenated sub-systems [1]:
$$H(s)=H_1(s)H_2(s)=\frac{1}{s+1}\tag{1}$$
with
$$H_1(s)=\frac{s-1}{s+1}\quad\textrm{and}\quad H_2(s)=\frac{1}{s-1}$$
Clearly, the total system is BIBO-stable because there are no poles on or to the right of the imaginary axis, but the system $H_2(s)$ is unstable, and hence, the total system is internally (asymptotically) unstable. This cannot be inferred from the transfer function $(1)$ because of the pole-zero cancellation. Only a state-variable description of the system could reveal its internal instability.
[1] B.P. Lathi and R. Green, Linear Systems and Signals, 3rd ed., p. 200.