The move was agreed at a regional summit in Chad, whose president described CAR as a "wound in the heart of central Africa".
"Armed bands loot, hold to ransom and racketeer the population," he said.
The former rebels had asked both their neighbours and former colonial power France to increase their forces.
The current 500-strong multinational peacekeeping force Fomac was "insufficient to do the job", Mr Deby said earlier, reports the AFP news agency.
He said the reinforcements would help restructure the security forces.
South Africa, which lost at least 13 troops during the rebel takeover, has said it is pulling its forces out of CAR.
South Africa's President Jacob Zuma attended the summit, along with central African leaders and a delegation from the CAR's new leaders.
The summit of the Economic Community  of Central African States (ECCAS) accepted former rebel head Michel Djotodia as the country's leader but that he should not be called president, Reuters news agency reports.
"Mr Djotodia will not be called president of the republic, but head of state of the transition," it quotes President Deby as saying.
Mr Djotodia was declared interim president last week and says he will hold elections within 18 months.
Earlier, the UN children's agency warned that increasing numbers of children are being wounded in the conflict.
Unicef's emergency co-ordinator in Bangui, Robert McCarthy, told the BBC's Focus on Africa programme that all armed groups should ensure that children were protected.
"We're calling for: Number one, the protection of children by all parties - by everyone with a gun, everyone in command of soldiers, we have to protect the children, equally women, vulnerable populations.
"Equally, we need security to resume the most basic of life-saving interventions for the aid providers in the city and especially in the countryside."
The Seleka rebels, an alliance of armed groups, seized power after a peace deal negotiated with ousted President Francois Bozize collapsed.
Mr Djotodia is a former civil servant who fell out with Mr Bozize and went into exile, before returning under the peace deal in January as defence minister.
He resigned in March before seizing power.
Correspondents say he does not have widespread international support and has fallen out with some of his former allies.
CAR has an unstable history and is extremely poor, though it has large deposits of minerals including gold and diamonds.
The Central African Republic's neighbours have agreed to send an extra 2,000 troops to help restore security following last month's rebel takeover.