Critics claimed Philip Hammond's financial statement was left in "disarray" after Conservatives called for reforms to National Insurance to be dropped.
Mr Johnson will appear at the despatch box when discussions continue in the House of Commons, despite little crossover between his international diplomacy brief and the Government's tax and spending plans.
Sources said the Cabinet minister was one of the Government's best communicators and a solid parliamentary performer.
The request was made before Mr Hammond delivered the Budget.
Prime Minister Theresa May promised to listen to concerns raised by Conservative MPs over the £2 billion hike in National Insurance contributions (NICs) for the self-employed and said there would be no vote until the autumn.
Labour claimed the promise amounted to a "partial U-turn" on the proposals set out by Mr Hammond.
But the Prime Minister insisted the planned 2% hike in Class 4 NICs was "fair" as benefits for self-employed workers have improved.
A review of modern employment practices by RSA chief executive Matthew Taylor, due to report over the summer, will be followed by a Government paper that is expected to include proposals to extend benefits such as parental leave to the self-employed.
Around 20 Tories have raised concerns about the changes, including Wales Minister Guto Bebb who said the Government should apologise for breaking an election manifesto commitment not to increase National Insurance.
Mrs May acknowledged on Thursday that the Budget had meant "difficult decisions", but insisted it was vital to close the gap between the amount of tax paid by the self-employed and those in "traditional" employment.
Mr Johnson will be responding to MPs during a section in the Budget debate on "Britain's place in the world".
Treasury minister David Gauke will sum up for the government.