The two companies have not disclosed how much Standard Life paid for the businesses, with various estimates ranging between £50m and £250m.
Elevate, which operates through independent financial advisers (IFAs), currently has 160,000 customers and manages £9.8bn of assets. The acquisition will see Standard Life's platform business grow to £36.4bn in assets under administration and 350,000 customers.
Standard Life said customers will benefit from the additional scale, expertise and resources the deal will bring, as well as discounts from leading UK fund managers.
Barry O'Dwyer, managing director of corporate, retail and wholesale at Standard Life, said: "We have a track record of successful acquisitions and an integration approach focused on building a sustainable commercial model for the Elevate platform."
The deal is part of AXA's wider decision to move out of life assurance and focus its UK activities on general insurance, health and asset management.
"AXA has concluded that, within the context of the UK Life and savings market, our business could have an even stronger future under the more focused ownership of specialists in their respective sectors," said Paul Riddell, head of strategic communications at AXA.
What does this mean for existing AXA customers?
When a new company takes over a platform this can mean there will be changes for its existing customers, as was the case when Alliance Trust Savings acquired Stocktrade.
Colin Low, chartered financial planner and managing director at Kingsfleet Wealth, said the deal had been driven by AXA's difficulties hitting financial performance targets on its platforms, in contrast to Standard Life whose wrap platform has been going from strength to strength.
But he said AXA Self Investor customers and Elevate advisers need to be aware of the different funds on offer across Standard Life and AXA's platforms, and the risk that costs could go up.
"There does appear to be some disparity," he said. "The Standard Life platform charges more than AXA Elevate. Our concern is that they may try to harmonise costs and raise charges. There seem to be hints of this with talk of commercial pricing, and I suspect that may spill over into the Self Investor product. It's going to take several months for it all to be pulled together, but we think it's something investors need to be aware of."
Standard Life's head of adviser and wealth manager propositions, David Tiller, said he could not comment on pricing now as this will be part of a review after the sale goes through. But he said Standard Life would aim to protect client interests and would also be consulting with advisers to find out their needs.
"The most important thing for us is making sure that it will not disrupt service for our advisers or AXA's Elevate advisers," he said. "We're keen to talk to them. I'm acutely aware that the latter selected Elevate not Standard Life, and they had good reasons for doing so, and we need to understand them."
Regarding what will happen to the AXA Self Investor platform, Mr Tiller said that the business represented a very small part of the deal, but that Standard Life plans to maintain continuity of service under the existing mechanisms.