When Darwin developed the theory of evolution by natural selection, he was at that time not able to identify the underlying mechanism responsible for the hereditary of traits. Of course, the hereditary of traits is something we can directly observe, and exploiting such a mechanism is exactly what a person breeding dogs, horses or roses does.
The discovery of genes and DNA revealed the underlying mechanism. This is an important point, because for scientists to be really satisfied you need both a verified observation and a plausible mechanism.
Since evolution occurs through the gradual change of traits, which is reflected by gradual changes in the genes, the theory of evolution makes a prediction: The closer two species are related in their evolutionary history, the more similarities in the genetic code we should expect. Molecular phylogenetics (i.e. inferring relatedness from DNA sequences) produces results that are consistent with other methods of inferring relatedness among species: morphology, geography, and the fossil record. Therefore, it adds credibility to the only theory that predicts these patterns, evolution from a common ancestor. It extends the theory of evolution by allowing inferences to be drawn even in the absence of geographic and fossil-based evidence.
While in science, one does not deal in absolute truths and proofs, but merely with hypotheses and validations or falsifications thereof, this discovery lends credibility to the theory of evolution.
Life only had to happen once, and from that single point of origin, all else diversified.
So yes, plants and animals indeed are hypothesized to share a common ancestor and DNA sequencing is continuing to generate more and more proof:
'29+ Evidences for Macroevolution'