Let me first mention that your problem is not symmetric and that if $M_1$ is an NFA and $M_2$ a DFA, then the inclusion problem is polynomial (simply because it amounts to test whether the complement of $L(M_2)$ intersects $L(M_1)$ ).
There is an important and non-trivial generalisation of this result: when $M_2$ is non-ambiguous (over any input, there is at most one accepting run). This result is proved in [SH85], but it also has an interesting connections to Schützenberger's work [S61], and as a consequence with NXA as mentionned in an answer above. (let me thank Jacques Sakarovitch for showing it to me).
Let me sketch a proof of this result.
Theorem Consider non-deterministic automata $M_1$, $M_2$, $M_2$ being non-ambiguous. One can decide $L(M_1)\subseteq L(M_2)$ in polynomial time.
Step 1: This reduces to universality of unambiguous automata.
Substep 1: we can assume $M_1$ to be deterministic (for this, it is sufficient to reformulate the problem over the alphabet consisting of the transitions of $M_1$).
Substep 2: then $L(M_1)\subseteq L(M_2)$ is equivalent to the universality of $L(M_2)\cup L(M_1)^c$, which is represented by a non-ambiguous automaton of polynomial size.
Step 2: It happens that unambiguous automata can be seen as NXA automata (Non-deterministic XOR automata in the previous post by RB) without there evaluation to be changed (indeed, a disjunction over all accpeting runs is equivalent to a xor over all accepting runs since there is at most one such run). For these automata, universality is known to be polynomial (QED).
Let me not that this polynomial universality result was known well before 2009.
Indeed, NXA automata are classically known as weighted automata over the field $Z/2Z$.
Such automata were introduced by Schützenberger in [S61]. He proves in this paper that the equivalence (and consequently universality) is decidable (using minimization) for all effective fields [S61]. The polynomial complexity some years after.
[SH85] Richard E. Stearns and Harry B. Hunt III. On the equivalence and containment prob-
lems for unambiguous regular expressions, regular grammars and finite automata.
SIAM J. Comput., 14(3):598–611, 1985.
[S61] Schützenberger, M.P.: On the definition of a family of automata. Information and
Control 4, 245–270 (1961)