The defence ministry said two Tornado jets hit the post with a Brimstone missile used against tanks and the vehicle with a 500-pound (230 kilogramme) Paveway IV laser-guided bomb.
"Both assessed successful," read one of the tweets.
Another said: "Tornado jets have carried out first air strikes in support of democratic Iraqi government."
It did not say when or where the strikes were carried out but explained they had aided Kurdish troops in the area.
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Fallon said the strikes were carried out by two jets from an RAF base in Cyprus.
"They identified and attacked a heavy weapon position that was endangering Kurdish forces and they subsequently attacked an ISIL armed pickup truck in the same area," he said, using an alternative name for the Islamic State group.
The British parliament last week approved a motion to join in a US-led military campaign against IS jihadists who have seized huge swathes of Iraq and Syria in recent months.
The government has said it will not send combat troops on the ground or join air strikes in Syria without further parliamentary approval.
British foreign minister Philip Hammond earlier on Tuesday said UK forces would not be "panicked" into dropping bombs in Iraq.
"When we do release our weapons we have to be absolutely sure that they are against ISIL targets, that they are not going to kill innocent Sunni Muslim civilians in areas that are occupied by ISIL," he told the BBC.
Asked about France joining the US-led coalition before Britain did, he said: "There is nobody who knows anything about air power who is suggesting that the French air force is a more formidable force than the RAF."
France has already carried out two rounds of air strikes.