The aircraft carrier will spearhead Carrier Strike Group 21 (CSG 21), the largest British peacetime deployment in a generation.
Hundreds of people lined the harbour walls to wave off the £3.2bn ship on Saturday as she set off on the 28-week deployment to the Indo-Pacific region which will cover 26,000 nautical miles.
Earlier, the Queen flew on board the vessel to wish the crew well.
She was welcomed by the ship's Commanding Officer Captain Angus Essenhigh and Commodore Stephen Moorhouse, Commander of the UK CSG, after arriving by helicopter while the 65,000-tonne ship was still at base.
She was given a briefing on the deployment which will take the CSG through the Mediterranean to the Red Sea, then from the Gulf of Aden to the Arabian Sea and Indian Ocean to the Philippine Sea.
The Queen also had a chance to meet some of the 1,700 personnel on board the carrier which includes Royal Navy sailors, Royal Air Force airmen and women, Royal Marines and 250 United States personnel.
Her hour-long visit followed a tour by Prime Minister Boris Johnson on Friday.
Watch: The Queen visited HMS Queen Elizabeth earlier on Saturday ahead of the ship setting sail.
The CSG will carry out visits to 40 countries including India, Japan, South Korea and Singapore with more than 70 engagements including sailing alongside the French carrier Charles De Gaulle in the Mediterranean.
Ahead of the ship's departure, the Defence Secretary said the deployment shows "we are strong on our own but even stronger with our allies".
Ben Wallace added: "The UK's Carrier Strike Group sets sail to write Britain’s name in the next chapter of history – a truly global Britain that steps forward to tackle the challenges of tomorrow, working hand-in-hand with our friends to defend our shared values and uphold the rules-based international order."
The deployment has been organised as part of the "UK's tilt to the Indo-Pacific region" in a bid to "bolster deep defence partnerships" as well as to take part in an exercise to mark the 50th anniversary of the Five Powers Defence Agreement with Malaysia, Singapore, Australia and New Zealand.
Eighteen F-35B aircraft are embarked on HMS Queen Elizabeth for the deployment – eight from the RAF's 617 Squadron, also known as the 'Dambusters', and 10 from the US Marines Corps.
The Britsh F-35Bs on HMS Queen Elizabeth will join operations against the so-called Islamic State in Iraq and Syria during the deployment.
Operating alongside the jets on CSG 21 are four Wildcat maritime attack helicopters, seven Merlin Mk2 anti-submarine helicopters and three Merlin Mk4 commando helicopters – the greatest quantity of helicopters assigned to a single UK Task Group in a decade.
Accompanying HMS Queen Elizabeth will be a surface fleet made up of Type 45 destroyers HMS Defender and HMS Diamond, Type 23 anti-submarine frigates HMS Kent and HMS Richmond, and the Royal Fleet Auxiliary's RFA Fort Victoria and RFA Tidespring, as well as the US destroyer USS The Sullivans and Dutch frigate HNLMS Evertsen.
A Royal Navy Astute-class submarine will also be deployed, armed with Tomahawk cruise missiles.
Commodore Steve Moorhouse, Commander United Kingdom Carrier Strike Group, said this marks "a new phase opens in Britain's maritime renaissance".
Watch: HMS Queen Elizabeth Leaves Portsmouth For Maiden Deployment
"HMS Queen Elizabeth, her escorts and her aircraft will now begin the most important peacetime deployment in a generation," Commodore Moorhouse said.
USS The Sullivans departed from Portsmouth just prior to HMS Queen Elizabeth setting sail.
The carrier had not been expected to return to Portsmouth after taking part in a major exercise off Scotland and the CSG had been expected to gather in the Solent prior to departure.
But heavy winds led to the unscheduled stop in the naval base with most of the other ships waiting at Devonport to regroup before sailing for the Mediterranean.