Since Devil's Sight allows one to see through magical and nonmagical darkness, and Silent Image can create any visual phenomenon, can I create darkness as a visual phenomenon which Devil's Sight can then see through?
Silent Image probably does not create the same effect as Darkness
Silent Image, a 1st-level spell says:
You create the image of an object, a creature, or some other visible phenomenon [...] The image appears at a spot within range and lasts for the duration. The image is purely visual; it isn't accompanied by sound, smell, or other sensory effects.
Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the creature can see through the image.
The wording of Silent Image is a little vague because it allows the caster a lot of room for creative use but it is probably unbalancing to allow it to duplicate the effect of the higher level Darkness spell. The fact that a creature can investigate it to nullify the effect means that it is not as strong as the true Darkness spell and so there is room to argue both ways.
I think it's also worth pointing out that the Silent Image spell does not convey any sort of effect or benefit to the generated image ("It is purely visual"). If you create the image of lava, it does not deal fire damage to any creature that stands in it. (That would be what Phantasmal Force, another higher level spell, does.) So it shouldn't convey the benefits of Darkness spell by the same token.
Even if you did rule that Silent Image replicated Darkness, in actual use, it might not function as well. The rule for Silent Image contains the sentence "Physical interaction with the image reveals it to be an illusion, because things can pass through it."
This creates a conflict: A creature or object can move through both a true Darkness and an illusory version just as easily. But the fact that the Silent Image version specifically states that "physical interaction reveals it to be an illusion" means that there must be some difference between the two (whatever that is) which causes it to not function as a true Darkness.
In the cleanest presentation, a creature might be outside of the effect and look through it at another creature that is also outside the effect. But as soon as, say, a projectile traveled through it, that would arguably be physical interaction with the illusory darkness and therefore reveal it to be normal darkness.
In short, it's messier to give Silent Image the ability to replicate all aspects of the Darkness spell than it is to say that it can create an area of darkness that lacks the full benefit of Darkness and therefore it's probably a better to interpret the rules so that it does not.
By the rules, we know that you can create an illusion of darkness, and I suppose even magical darkness. The rules are not entirely clear, but by logic there are only two possible ways a DM could rule on this:
Interpretation #1: Yes.
Since you copy magical darkness so perfectly, the Warlock's Devil's Sight would work, and they see through it. Even if they don't do a great job, it would still be darkness, and the Devil's Sight should work on it. It seems reasonable that any darkness effect would be covered by the term:
You can see normally in darkness, both magical and nonmagical, to a distance of 120 ft
Interpretation #2: No.
A weaker argument could be made that because the illusionist doesn't know about/can't copy the magical nature correctly just create "blackness" that isn't really magical darkness. Since it doesn't work like magical darkness, the Warlock not being able to see through it would have reason to believe something odd is going on and would have a reason to investigate to learn if it is an illusion.
Odd Edge Case
Note: there is (at least) one form of darkness that Crawford tweeted Devil's Sight is known not to work on, and that is the void created by hunger of Hadar. One small issue to that is that I don't think it made it into Sage Advice, and so despite being official ruling when it was made, it wouldn't be under the new Sage Advice header. Take all of that for what you will, but an illusion of the void, would be harder to rule.