When the valiant handyperson manages to get machine grease on the rug while fixing the clothes dryer, like I did today, the immediate solution is to dampen a clean rag with hot water, put some concentrated liquid laundry detergent directly on the rag and work it into the grease. Clean the rag and do it over until the grease is gone. Then clean the rag, wet it and scrub-rinse the area several times with just clear water.
Your significant other will never guess. Then again, if that doesn't work -- go to the store and get some Oil Lift. It rarely bothers rug colour (but test in the corner first anyway) and always removes any oil based mess.