The answer is almost yes if you mean in your post the following:
$C^1$-topology should mean the compact open $C^1$-topology and you should replace the $d$ by the mapping

$$T \colon C^1 (M,N) \rightarrow C(M,N) \times C (TM,TN), f \mapsto (f,df)$$
where the factors on the right are endowed with the compact open topology.
Then the compact open $C^1$ topology can be defined as the initial topology with respect to $T$ (note this is equivalent to stating that $T$ is a topological embedding onto its closed image (which is what you asked)).
Details of this construction can be found in Wockels script https://www.math.uni-hamburg.de/home/wockel/teaching/data/HigherStructures2013/hs.pdf
in Section 5 (note that he deals immediately with the compact open $C^\infty$ topology and all proofs deal with this more complicated situation, however all the proofs adapt to switching from $k=\infty$ to $k=1$.

Some comments: For many desirable properties of the compact open topology one would like $M$ to be compact (see Wockel's notes).
If $M$ is non-compact there are also different function space topologies, often called Whitney topologies which are often considered and which coincide with the compact open $C^1$-topology if $M$ is compact.