To demonstrate this handy trick, I worked up a very simple photo tutorial. You'll only have to look at it a couple times and work it thru a couple times to get it and love it and never go back to sewing things together ever again! So here we go.... check this out!
Insert your hook thru both sides of the piece that you are attaching and the st of the piece that you are working up (1), pull the yarn through all of those sts (2) and finish your single crochet (3). Repeat all 3 steps until the piece is completely attached.