2071 pieces scattered across a gallery floor can make you want to give up before you have even picked up the first one. Bad idea. A jigsaw, in real life as in the game, is won through method, never through panic. Here is how to approach Cleaning Up The Puzzle Gallery without ending up buried.

The idea is simple: the gallery has 87 empty frames, one per painting to rebuild, and a heap of pieces spread across the rooms. Your job as janitor is to return each piece to its artwork. So the very first reflex to adopt is to never think "2071 pieces." Think "one painting at a time." One frame, its pieces, done, move on to the next. It is the only way to keep a cool head.
The temptation, when everything is scattered, is to scoop it all up at once. Resist. Pick a painting, ideally a small format or a work with very sharp colors to start, and focus only on its pieces. Every painting in the game is visually distinct, thirteen different artists from Monet to Hokusai, which helps enormously to know which piece belongs to what. A Van Gogh sky does not look like a Hokusai wave: use that.
No magic here, just the good habits every puzzler knows. Start with the edges and corners: they are the easiest pieces to identify, with their flat side, and they give you the frame everything else will slot into. Then sort by color zones and patterns before assembling. Gather all the sky blues, all the greens of a foliage, all the pieces with a recognizable texture. Sorting is 80% of the work. Once the pieces are grouped, assembly flows on its own.
Keep an eye on the oddly shaped pieces too. Unusual cuts are your best friends: they only have one possible spot, so set them aside and place them first once their zone takes shape.
The game promises enhancements to back up your puzzling skills. Do not hesitate to use them. We are not here to punish ourselves, but for the quiet pleasure of watching a masterpiece come back to life. If a feature helps you sort, spot or highlight pieces, take it, especially on the trickier paintings or the big monochrome zones that are the real bane of any jigsaw.
One last tip, maybe the most important: do not play this like a race. Cleaning Up The Puzzle Gallery is a cozy game, with its classical music and its calm museum mood. Put one session into an evening, finish a painting or two, and come back the next day with a fresh mind. That is how you get through a giant puzzle without hating it. At home as in the game.