If you're looking for Lovecraftian influenced anime let me recommend the series Demonbane (2004).
Set in Lovecraft’s own witch haunted city of Arkham in the 1920s, Demonbane follows detective Kuro Daijuji whose life changes forever when he meets a mysterious girl named Al Azif, who turns out to be the living personification of Lovecraft's dreaded Necronomicon. Al (as she's called) is being hunted down by the nefarious occult group The Black Lodge who want to use her for their own diabolical purposes, so she consigns herself to Kuro and makes him into her master (even though he knows next to nothing about magic).
The first agent The Black Lodge sends after them is (a guitar playing) Dr. Herbert West. A magic duel breaks out between Kuro and West and West ends up summoning a giant robot called a Deus Machina. Al tries to summon her Deus Machina, Aeon, but is unable to, instead locating one called Demonbane. The two make due and pilot Demonbane into battle and win.
It turns out that Demonbane belongs to the Haddo Organization which is attempting to thwart the schemes of The Black Lodge. To this end Kuro and Al becomes agents of Haddo Organization.
Eventually The Black Lodge starts sending more agents after Kuro and Al, each of which have their own Deus Machina. So yeah this is basically a mecha anime.
Aside from the story being set in Arkham, Al being the Necronomicon and Dr. West being a recurring character, other Lovecraft related elements include an appearance by just about every major grimoire featured in the Mythos, some of them personified as girls like Al, some just as books. Some of the Deus Machina are named after Mythos creatures; one is called Byaknee for example. The Powder of Ibn Ghazi is used to make magically infused bullets. Episode 5 is based around "The Shadow Over Innsmouth" (in fact that's the name of that episode) and at the end of that episode Kuro and Al have to use Demonbane to fight Dagon himself. Other Great Old Ones who actually appear include: Cthugha, Ithaqua, Nyarlathotep (who of course is always scheming), and of course The Big C himself. There are some other Lovecraft allusions but to talk about those would be to give away certain plot twists.
Also I've read that Kuro's name is a reference to British Cthulhu Mythos author Brian Lumley's paranormal investigator character Titus Crow, because "Kuro" is the Japanese word for "black" and Crows are black birds, so....
Like Nyaruko-san: Another Crawling Chaos, Demonbane's entire 12 episode run (it's not a long series) can be watched on Crunchyroll.com in Japanese with English subtitles. Currently there is no American DVD release.
Also look out for the work of Japanese horror writer Chiaki J. Konaka who is known for infusing his projects with Lovecraftian touches.
He was the head writer on the mecha anime The BIG O (1999-2003) which is set in the future following an apocalyptic fallout which has forced western civilization to revert back to the 1920s, albeit with certain technological advantages such as android maids and, of course, giant robots. Robot battles aside The BIG O mostly involves people digging into the past trying to uncover what happened to cause the apocalyptic fallout that forced human civilization to end up in the present state it is now. Of course as in any Lovecraft story digging up the past never turns out well. In particular however check out The BIG O Season 1 Episode 7 “The Call from the Past” which contains a number of shout-outs to Lovecraft’s “The Shadow Over Innsmouth” including an aquatic enemy mecha called Dagon.
Konaka also worked on Toei Animation’s popular dueling monsters TV series Digimon, originally as a scenarist on the second season writing two major side-story arches, one of which involved the child protagonists trip to the “dark digital ocean” where they are threatened by the Deep One-like Scubamon and the Cthulhu-shaped Dagomon who they serve.
Konaka later became the head writer for the third season of Digimon, known as Digimon Tamers (2001). As Anime News Network critic Hope Chapman notes, due to Konaka’s Lovecraft fueled influence Digimon Tamers is by far the most frightening, and at certain points disturbing, season of Digimon ever produced. The central antagonist of the series, a sentient computer virus called the D-Reaper, certainly owes a lot to Lovecraft’s Great Old Ones in both conception and execution. In addition to this there are Lovecraft related references peppered throughout Digimon Tamers including references to Miskatonic University and a mysterious organization called Hypnos (from Lovecraft’s 1922 short story of the same name) which commands a pair of weapons called Yuggoth (from Lovecraft’s “The Whisperer in Darkness”) and Shaggi (from Ramsey Campbell's 1964 short story "The Insects From Shaggai" based on an idea of Lovecraft’s).
Digimon Tamers is my favorite season of Digimon for many reasons, not just the Lovecraft references. An English dub of it aired on Fox Kids in the early 2000s but is currently unavailable on DVD. You can watch the entire series on Hulu however in Japanese with English subtitles, which I recommend anyway.
Lastly, even though it's not anime, let me just mention the tokusatsu series Ultraman Tiga (1996), part of Japan's long running (since 1966) Ultraman series. The last three episodes of the series were written by Konaka who based them upon Lovecraft's "The Call of Cthulhu" and "Out of the Aeons". Basically R'lyeh rises from the deep and darkness consumes the earth. Ultraman has to go to the island and fight the Elder God Ghatanothoa (from "Out of the Aeons") and save the world. It's great. Ultraman Tiga was dubbed and broadcasted on Fox Kids in the early 2000s and is available on DVD in the United States in Japanese with English subtitles.