She had faced growing pressure to resign amid claims she misled parliament over targets for removing illegal migrants.
Environment Secretary Mr Gove insisted he still has confidence in Mrs May who has suffered a series of setbacks following the publication of her draft Brexit deal with Brussels.
Sky News' political correspondent Tom Rayner said Ms Rudd has risen "Lazarus-like" and pointed out that she is "not just someone who supported Remain - she was a vocal campaigner during the referendum but also just in the last few weeks she has said she thinks that a People's Vote would be preferable to no deal".
Mrs May agreed a draft withdrawal agreement for Brexit with her cabinet on Wednesday, which had already been signed off by negotiators from both the United Kingdom and EU.
Ms Rudd described herself as a "one nation Tory" and added that she is "delighted to have been given the honour and the responsibility" of heading up the department for work and pensions.
"After the Home Secretary had given her answer in the hearing, there were confused email exchanges trying to establish the position on targets".
Labour MPs Diane Abbott and David Lammy, who helped to expose the Windrush scandal, slammed Ms Rudd's appointment tonight.
The new Minister for "breccia" the Prime Minister has appointed Stephen Barclay, after in protest the Cabinet left Dominic Raab.
Ms Rudd said she had seen universal credit "transform lives" in her Hastings and Rye constituency but she "recognised there had been some issues with it".
She said she would make it her role to to "iron out those difficulties and make it a force wholly for good".
But a report into the row found officials had repeatedly given her the wrong information and then failed to clear up the problem so that she could correct the record. A blow to the First minister, increasingly isolated since she won on Wednesday a draft agreement with the European Union on the Brexit.
Her return to the frontbench was backed by fellow Cabinet members Sajid Javid and Liz Truss.