Her beaming face is the first thing you see at Tate Modern – Swiss artist Sophie Taeuber-Arp (1889-1943) swaying sideways in a solo dance performance, wearing an op-art dress at the beach or grabbing the camera’s attention as the only woman in a crowd of dedicated Dadaists. His smile comes out every time. And the fact that most of us barely recognize this prodigiously gifted artist seems to maximize the surprise of all that follows, from sparkling paintings and sculptures to graphics, through embroidery and humorous puppets. Taeuber-Arp is the great little-known modernist.

The reasons are not beyond reach. Sophie Taeuber was born in Davos and learned to sew from her mother, a young thrifty widow. She chose a training in practical arts and crafts rather than the usual fine arts education, so that she could turn wood and make supports, warp a loom and solder silver where other artists were learning. to draw from old plaster casts. She studied with modern dance pioneer Rudolf von Laban, designed glittering posters and created elegant tongue-and-groove furniture. But so little of that survives: the performances that have never been filmed, the objects that have disappeared, the paper that has naturally disappeared.

Then she married the much more famous Jean (Hans) Arp in 1922, and became deeply involved in the playful and anarchic activities of the Dadaists. Husband and wife made many mutual works of art. They also traveled to Europe, possibly in an attempt to avoid the Nazis. And since the paper alone was light enough to be carried through last minute border crossings, Taeuber-Arp returned to drawing. In a very moving room at Tate Modern, you can see the works that have survived – their lines fluttering, dancing and crisscrossing in exquisite abstract formations.

In January 1943, after fleeing to Zurich, she missed the last tram home one night and slept in a snow-covered summer house. The stove failed to ignite and she was found the next day, dead from carbon monoxide poisoning. Unlike her husband, whose biomorphic sculptures enjoyed intense success in the post-war era, Taeuber-Arp was forgotten for the next half-century.

To see his work at Tate Modern is to be immediately elated. The show opens with small watercolors that play luminous rectangles against triangles, squares and disks. They look like the art of their time; think of Malevich, Mondrian or Klee. And Taeuber-Arp is an abstract artist from first to last. But these enchanting paintings always carry humorous hints of reality – a quirky face, a tube of lipstick, puffs of summer perfume, funny boats and knights on horseback.

His paint colors are delicate like a Klee, sometimes combined with gold or silver leaf. And then, as your eye gets used to this painted art, the medium suddenly changes. What looks like watercolor is actually wool, which somehow glows as radiant as the golden frame that contains its pristine geometry. How is it, you ask yourself, scrutinizing this needlework of 1918? Taeuber-Arp knew exactly how to choreograph each point, playing them in different directions to catch the light and generate the glow.

It used to be called the applied arts (and still is in some patronizing circles). But Taeuber-Arp believed that beauty should be useful, and vice versa. She is with William Morris in this regard. Among the most breathtaking works is a sequence of beaded bags, whose surfaces are a fascinating combination of scintillating light and glassy opacity, constructed to hang like mobiles from threads of jewelry. Sure, you can imagine them swinging from a woman’s wrist at a party, but in this context, they look like they were to the artist: weird new sculptures.

One of the earliest cushion covers has been so perfectly preserved that it looks like no one could have leaned on it. This needlepoint tapestry invents abstract motifs in endlessly renewed sequences (including a touching square of trial and error that the artist has never repeated) which seem even richer than the paintings. The colors are surprising: shocking pink and lime green, maintained in a perfect tonal balance. Where the hell did she get such dyes in 1916? One answer is that the artist had very wealthy patrons. Taeuber-Arp may have been overlooked since her death, but she was deservedly successful during her lifetime.

In Strasbourg, she designed the Five O’Clock tea room and the elaborate bar at the Aubette, all rectangles and points. The collector André Horn commissioned stained glass windows for his large apartment which prefigure the minimalism of the 1960s. Taeuber-Arp also designed a house for her and Arp on the outskirts of Paris: an ocher cube that merged high modernism with the stone walls of an alpine farmhouse.

The paintings she creates there must be severely abstract, given their rectilinear shapes and restricted colors – often three, sometimes only two – and yet they are wonderfully exuberant. Animated circles (1934) plays rows of blue records against a white plane with all the quirky glee of a bowling alley in warm summer shadows. It is an agile art: athletic, calm, demanding, always on point.

The admiration for art extends (for me) throughout professional life. Taeuber-Arp taught for many years, commuting daily to Zurich by train. She performed as a dancer, made puppets for dada theatrical productions, edited an art magazine, wrote art books, had a parallel career as an architect and worked in almost all artistic media. She signed her tapestries, sold her jewelry necklaces to art collectors, made no distinction between top and bottom, applied art and fine art, like her modernist colleagues.

And there is modesty in his inventions. Towards the end of this exhibition hangs a painting composed, it seems, entirely of disks – red, white, blue – on a pale background. They twinkle across the room, flashing happily in the eyes. But as you get closer, a mysterious blur occurs. These circles are actually the flat fronts of solid cones that recede into a dark wood structure behind. You move, the image moves. Three-dimensional painting.

Dada poet Hugo Ball compared Taeuber-Arp to a lark, taking the sky with her as she soared happily upward. And it seems that everything she learned in her youth has been picked up and transferred into her art – from the precision of woodworking and sewing to the abstraction of needlepoint grids. She wasted nothing; and the tragedy of his accidental death is fully apparent at Tate Modern, as the show abruptly ends. Without a doubt, Taeuber-Arp would have continued to invent elegant and beautiful objects, sending them out with his stated goal of enlightening this world.