* ASML beats Q4 sales, earnings expectations

* Says 2021 order book looks strong

* Forecasts 4 billion euros in sales in Q1 2021

* Could beat 2021 'low double digit sales growth' outlook

* Shares extend recent rally to hit record high

AMSTERDAM, Jan 20 (Reuters) - ASML, which supplies equipment that helps semiconductor makers create the circuitry of computer chips, outstripped fourth quarter earnings forecasts on Wednesday, driving its shares to an all-time high.

The Dutch company is hoping to be able to begin shipping its newest and most expensive tools to Chinese semiconductor manufacturers after a U.S. campaign by the Trump administration hampered it from doing so.

It said fourth quarter net profit was 1.4 billion euros ($1.7 billion) on sales of 4.3 billion euros, compared to 1.13 billion euros and 4.04 billion euros a year earlier.

Analysts polled by Refinitiv had forecast profit of 1.03 billion euros at ASML on sales of 3.73 billion euros.

ASML has a near-monopoly on lithography machines, which its CEO Peter Wennink has likened to 200 million euro photocopiers used by cutting-edge chipmakers including Taiwan Semiconductor, Samsung and Intel to create chips in devices ranging from consumer electronics to cars to artificial intelligence.

Wennink said ASML had exceeded forecasts because of additional shipments of its mid-level "Deep Ultraviolet" lithography machines and upgrades to existing systems, with high utilization of chipmaking factories.

Shares in ASML, which have rallied in recent weeks on news of chip shortages in some sectors and on plans by key customers to invest heavily in new plants that require ASML equipment, rose 2.3% to an all-time high of 450 euros at 0827 GMT.

ASML forecast 4 billion euros in sales for the first quarter of 2021 on bookings of 4.2 billion euros. In November it had said that it expected "low double digit" sales growth in 2021.

Citi analyst Amit Harchandani, who rates ASML shares a Buy, said the modest share rise was due to the "high expectations bar" the company faces, adding that the latest numbers would not lead to higher consensus for full year 2021 earnings.

Finance Director Roger Dassen said ASML expects 10% growth in machines for logic chip manufacturers, 20% growth in machines for memory chip makers and 30% growth in service revenue.

Those forecasts were "under the assumption that we will not be faced with more limitations on the geopolitical front and ... export controls than what we're seeing today," he said, adding they could be exceeded if controls were loosened.

So far, the Dutch government has withheld export licenses from ASML for shipping its most advanced equipment to mainland Chinese customers in the face of U.S government pressure. ($1 = 0.8235 euros) (Reporting by Toby Sterling; Editing by Himani Sarkar, Keith Weir and Alexander Smith)