A cheap, quick and easy wipe-free alternative is an oil cleanse. I was recommended this by a friend and was a bit skeptical but I can honestly say that for me (combination to dry (depending on hormones) sensitive skin) it has been amazing. It removes makeup faster and easier than most cleansers ive ever used - even removes waterproof mascara and doesnt sting, tingle or make things feel tight / raw.
You can use any oil (baby oil, olive oil etc) but a natural non-comedogenic one will work best for long term use (I.e. grapeseed, jojoba, peach kernal etc) and you can add in essential oils or use a pre-blended aromatherapy oil if you wanna get fancy. The 'best' way is to rub the oil into damp skin with circular motions and then remove with damp cotton pads or a flannel.
However my lazy / DIY wipe way also works: Apply a little oil to damp cotton pads and rub all over face, you may need two or three sets for a full face. Use a final set of cotton pads to remove excess oil - especially from the eyelashes as it can make things a little blurry until you have blinked it away. Done. Toning and moisturising are still beneficial but if you really want to just be lazy you can leave it without worry about whatever chemicals may have been smeared all over your face! I find I get a much deeper / better cleanse this way than using wipes too - no accidental panda eyes in the morning!
Although rubbing oil onto your face sounds like it will give you spots the extra oil helps to break down sebum as it is a similar consistency (more true with oils that mimic our natural sebum like jojoba) and often an overproduction of oil on the face is often due to the skin being dry and the body over-compensating - which this also addresses!