How deeply does the drain hose from the washer into the utility sink?
Before I switched over to an automatic laundry pump on my utility sink, I thought it was amazing that the basin never overflowed (my sink is below the drains from the house, as there had previously been a septic tank, and the tied through the septic system to connect to the sewer system)
I just thought it was a feature ... I'd just do like you, and run to spin the rest of the water out ... until I realized that it filled the water level just to the height of the drain hose. What was actually happening is the water was flowing from the drain hose back into the washer.
(mind you, the drain pipe had a U-bend in it, but I guess when the water was being ejected, it pushed the air out of the bend, so it could then act as a syphon ... or maybe it was just the difference in height between the full sink and the exit drain on the laundry that gave it enough pressure to backflow)
I've now got an automatic pump, but if it starts getting too much lint, down the drain, it'd start to cycle too often which would allow a fair amount of water to build up ... so I'd have to restrict the exit on the laundry pump, until I restricted it far enough that about every year or so I have to take it off and clean it.
They have some wire net things that you can attach to the drain hose, but I find they fill up too quickly, and they have large enough openings that they still let a fair bit through. I've switched over to getting knee-high stockings, and using that instead. (finer mesh and holds more so you don't have to change it out as often)
So anyway, I'd check to see how fast the utility sink is draining. It'd probably be good to check on it a few times while you run a load of laundry. If the water ever gets up to the level of the drain pipe from the washer, there's your problem.
If you can't get the sink to drain faster, you can always shorten how far the drain pipe extends into the sink.