You don't have to be the world's greatest detective to know that the BBC is not alone in updating Sherlock Holmes, but it might help. Elementary, the US version, has been largely ignored here, and it doesn't take strong sleuthing powers to see why. How dare they try to beat us at our own game – and how dare they change the gender of dear old Doctor Watson? Clearly, they haven't a clue. Or have they?

As is often the case, such ire is unfounded. Elementary, now into its second series, having launched on CBS in 2012, is rather good and often great. That's not something that was said much about the BBC's much-hyped Sherlock, starring Benedict Cumberbatch. For starters, there's Jonny Lee Miller giving a fantastic, career-best performance as Holmes, completely disappearing into the role. His high-functioning detective adds scruffy and twitchy to the usual roll call of the Holmes's attributes (focused, intelligent, arrogant, etc). Daringly, the show, which airs here on Sky Living, also gives Holmes a sex life: he frequently indulges in very casual encounters, which certainly makes a nice change. And, despite the New York setting, Holmes is still very much a Brit: it's wonderful to hear words such as twaddle, slappers and bollocks uttered so casually in a US show.

He's also a recovering drug addict, which is where Watson comes in. Played superbly by Lucy Liu, Watson has given up her career as a surgeon to become a sober companion for recovering addicts. She has been hired by Holmes's offscreen father to spend six weeks with the detective, living in his house, going everywhere with him in case he relapses. She starts off as an irritant to the great man – but, no big surprise, they gradually bond, bringing out the best in each other. As a double act, the two have great chemistry – but there's no sexual tension. In fact, the Beeb's Holmes and Watson are more likely to jump into bed together than these two.

We have to wait until episode 16 for them to become a proper crimefighting team, but then they're off, solving such posers as a plane crash where one of the victims died before hitting the ground. They find missing people, catch devious murderers and crack an uncrackable safe. Even when Elementary's mysteries aren't all that mysterious, the duo's interplay and charm carry things along. With each show a tight 42 minutes, the pacing feels perfect and the details are light: viewers can keep up with the investigation and feel involved, not something every investigative show achieves.

The addiction is respectfully handled. Holmes may put himself into a kind of trance during his group meetings so that he can sort through clues, but he comes to see the value of them. They play into the plots in smart ways, such as when an old drug-dealing pal, played by the always entertaining John Hannah, enlists Holmes's help to find his kidnapped daughter – and tries to get the sleuth high on coke, as he believes he's more effective and imaginative that way.

While it doesn't adhere to the letter of Arthur Conan Doyle's creation, it frequently captures the spirit. And familiar Holmes elements, such as arch foe Moriarty and femme fatale Irene Adler, are gradually added to the mix and given an interesting spin. But be warned: if the thought of a female Watson makes your blood boil, you'll probably explode when you see what they've done to nice old housekeeper Mrs Hudson.