Speaking on BBC Radio 4's World At One programme, General Lord Dannatt said it was wrong to "talk up" the incidents.
Five Afghan soldiers died in an assault on the former UK base Camp Bastion and capital city Kabul has been attacked.
Lord Dannatt said the Afghan government must retain the support of the West.
The last UK combat troops left Afghanistan in October, with Camp Bastion in Helmand Province being handed over to local armed forces.
But there has been a surge in insurgent attacks over the past month, targeting Afghan forces, foreigners and civilians. It has raised concerns about whether Afghan security forces will be able to protect the country when the bulk of the other foreign troops depart this month.
Three South Africans were killed in a Taliban attack on a compound used by a US-based charity on Saturday in Kabul, the third such incident on foreigners in 10 days.
Ahead of a conference this week on Afghanistan's future in London, Lord Dannatt said: "It is important the new government shows it has competence and confidence to govern... it is absolutely critical to show international determination to go on financing and supporting the developing Afghanistan."
He said: "The security situation is an important element in this but don't let's fall into the trap of playing the Taliban's game and talking up the number of attacks...
"Of course there are going to be attacks... a low level, even a medium level degree of insecurity and a continuing insurgency is quite likely, but let's keep the wider picture in mind."
The Ministry of Defence said Afghan security forces are providing effective security and the recent incidents do not represent the situation across the country.
Lord Dannatt's predecessor, General Sir Mike Jackson, has reiterated his view that the UK withdrew from Afghanistan too soon.
Interviewed on the BBC Radio 4's Today programme, he said: "The war, conflict, call it what you will, wasn't over. I'm left feeling there was no victory despite what the politicians say."
But he said that while the recent attacks were "somewhat depressing... I don't see certainly at the moment, and I hope never, that the Taliban will be in a position where they can overthrow the elected government of Afghanistan".
He said "great effort" has been put into training and mentoring the Afghan security forces.
Sir Mike, who led the Army between 2003 and 2006 at the height of the UK's involvement in Afghanistan, added: "Afghanistan is not a military problem per se, the root of it is political. It seems to me there can only be, in the longer term, a political outcome."
Recent attacks by the Taliban in Afghanistan are not a sign it is making advances as the last foreign troops prepare to withdraw, the former head of the British army has said.