Jones was on holiday in France when he heard the news of the invasion of the Falklands. He had just returned from training in Kenya, and his battalion, 2nd Battalion The Parachute Regiment, was earmarked for deployment for Belize. Jones' battalion was attached, alongside 3 PARA, to reinforce 3 Commando Brigade Royal Marines, the first major infantry formation to be sent south. During the Battle of Goose Green, an attack against entrenched Argentinian positions, with his unit pinned down by heavy fire from MAG machine guns and FAL automatic rifles, he led a charge against the nearest position. He was killed while doing so but the Argentinian unit surrendered shortly afterwards. For his actions he was posthumously awarded the Victoria Cross. Command of 2 PARA passed to Major Chris Keeble. Jones' body was buried in a battlefield grave at Ajax Bay on 30 May. After the war it was exhumed and re-buried at the Blue Beach War Cemetery in San Carlos on 25 October 1982. Ex-TA Para officer and military theorist Spencer Fitz-Gibbon wrote in 1995 that despite his undoubted courage H. did more to hinder than to help 2 Para, losing sight of the overall battle picture and failing to allow his sub-unit commanders to exercise Mission Command, before his fatal attempt to lead "A" Company forward from the position where they had become bogged down. Margaret Thatcher said "his life was lost, but his death was the turning point in the battle." The battle demonstrated the UK's increasingly unquestionable military superiority, quelling concerns about possible defeat, and led to the release of 112 civilians who had been imprisoned in the local community hall for the best part of a month.
Jones' grave at Blue Beach War Cemetery is marked with a headstone engraved with the Parachute Regiment's insignia and that of the Victoria Cross. The headstone includes the quotation: "He is not the beginning but the continuing of the same unto the end." A street in Stanley was named H Jones Road in his memory in addition to Jones Avenue in Mount Pleasant air base. A memorial stone to all those killed at the scene of the battle, near Darwin, also bears his name. His name is also on the South Atlantic Task Force Memorial in St Paul's Cathedral, London, on the wall with the names of the fallen in the Falklands Memorial Chapel at Pangbourne College, and the Parachute Regiment Memorial at their headquarters in Aldershot; he also has a memorial in the cloisters of Eton College and a plaque on a footpath at Kingswear, Devon. The memorial board from St Peter's School, carved with the name of Jones can be seen in Seaford Museum. In addition the 'Colonel H' Public house in Great Yarmouth, Norfolk is named in his honour. There is a wooden plaque memorial in Kingswear parish church and a copy of the citation is on view near the memorial.