You know what's better than a band listing their songs by Roman numerals on their album sleeve? A band who looked as if they'd broken up years ago listing their songs by Roman numerals on their album sleeve, that's what.

By chance, that's what Let's Stay Friends by Les Savy Fav is. Having spent six long years waiting patiently for Les Savy Fav to release an album of new material since 2001's Go Forth, the pressure must have been huge on the band to deliver something new that lived up to their proud, fiercely independent blueprint and satisfied their slavering fans at the same time. And we'll be buggered if Let's Be Friends isn't exactly that.

Outside of Spoon, American indie hasn't really offered us anything of worth at all this year. But we never gave up hope that something great would come along. We're just quietly staggered that it was Les Savy Fav who managed to come up with it. Pre-Let's Stay Friends, people liked Les Savy Fav because they'd seen them live – there's just something about a giant beardy man in his underwear hurling confectionery accompanied by a deafening punk roar that sticks in the mind. If Les Savy Fav could just harness that aggressive imagination in the studio, you suspected, they'd take over the world.

In that case, say hello to Let's Stay Friends – an album that refuses to let you even think as it hammers you dizzy; not just with the frayed-edge anger that you were expecting, but with moments of restraint. Yes, restraint. No, we weren't expecting it either.

Let's Stay Friends' mission statement is laid out – if not on its 'Adam and Eve being attacked by wild animals' sleeve – in its very first song, Pots & Pans. “There was a band, called the pots and pans," it goes. "And when they toured across the land, people said ‘No no no’ but the drummer said 'Yes yes yes'” Everything that follows is Les Savy Fav daring the world to do that to them again.

Be it screamed lead single The Equestrian – the one with the video of the toddler singing along that you saw a couple of weeks ago – or the !!!-ish funk of Patty Lee, the majority of Let's Stay Friends is whip-tight and focused on the finishing line like an Olympian. Barring a couple of songs so generic that you barely even notice they're there. Let's Stay Friends by Les Savy Fav is a heartstopping comeback of grin-inducing proportion.

Now never leave us again.