It surprises people, but a modern dishwasher run sensibly often uses less hot water and energy than washing the same dishes by hand under a running tap. The machine is efficient by design; the waste, where it exists, is in how it is loaded and which programme is chosen.
Why the machine can win
A dishwasher heats a precise, small amount of water and recirculates it, whereas a tap left running while you wash and rinse pours hot water away continuously. Filled properly and run on the right programme, the machine does a full load of dishes on less hot water than the sink method uses, and the energy to heat that water is the main cost either way. The hand-wash only wins if you are frugal with a bowl rather than a running tap.
Use eco mode and full loads
The single biggest saving is the eco programme. It washes at a lower temperature over a longer time, which uses noticeably less energy than the hotter, faster cycles, and still cleans normal dishes perfectly well. Just as important, run the machine only when it is full, since a half-empty load uses nearly as much water and energy as a full one. Waiting for a full load and choosing eco together cut the cost per item substantially.
Skip the pre-rinse
Rinsing plates under a hot tap before loading wastes exactly the hot water the machine is meant to save you, and modern dishwashers and detergents cope fine with scraped, unrinsed plates. Scrape the leftovers into the bin or compost, load them as they are, and let the machine do its job. The pre-rinse habit is a hangover from older, weaker machines and quietly undoes the dishwasher's efficiency.
A little maintenance
Keep the filter clean and the spray arms clear so the machine works effectively and does not need rewashing, and use rinse aid so dishes dry without a second hot cycle. If your machine can draw from a hot feed and you heat water cheaply with gas, that can help, though many are cold-fill and heat efficiently themselves. Run the wattage and cycle time through the cost calculator to see what an eco load really costs you.