In his legal advice on Tuesday morning, Mr Cox said the legally-binding changes in the joint instrument secured by Mrs May "reduce the risk that the United Kingdom could be indefinitely and involuntarily detained within the Protocol's provisions".
The EU, represented by its Brexit negotiator Michel Barnier, confirmed the EU had agreed to a "legally binding interpretation" of the withdrawal agreement that includes the option for the United Kingdom to leave the customs union unilaterally.
On Monday night the Prime Minister said that she has secured "legally binding" changes to the Brexit deal, which ensure the Irish backstop can not be permanent.
In a personal blow to the embattled leader, MPs rejected her deal by 391 votes to 242 after 24 hours of frantic diplomacy and a wildly fluctuating pound.
Ms. May wasn't helped by Attorney-General Geoffrey Cox, who released a legal opinion of the revised deal on Tuesday that indicated the changes she negotiated had limited impact.
There are three component parts of the new assurances for MPs, but none actually reopen the withdrawal agreement or remove the backstop mechanism.
"Brexit has been a dark cloud over us for many months, and particularly the threat of no deal", Mr Varadkar said in a statement.
"MPs were clear that legal changes were needed to the backstop", May said in Strasbourg.
She told lawmakers after the passed the non-binding motion rejecting a no-deal Brexit they are down to two choices: approving a withdrawal agreement in coming days and asking the European Union for a short delay, or requesting a "much longer" extension in hopes of negotiating a new arrangement.
Prominent Tory backbencher Damian Collins, who backed remaining in the European Union in 2016, said he will be voting against the Government's motion, while former Brexit minister and ERG member Steve Baker said he thinks the deal "falls very far short of what the Government whipped us to vote for".
After two-and-a-half years of haggling with Britain over Brexit, Juncker cautioned that this was the last chance for Britain.
Loyal ministers concede the deal is not ideal but say it is the best way to move forward - and that rejecting it could put Brexit at risk.
With the approaching deadline intensifying fears that economic and personal turmoil might follow a "no-deal" withdrawal by Britain, Parliament voted 321-278 Wednesday to rule out the possibility.
"If you pushed me to the end point where it's a choice between no deal and no Brexit".
Both Keir Starmer, the opposition Labour Party's Brexit spokesman, and Conservative lawmaker Dominic Grieve expressed skepticism about whether May had won substantive concessions.
MPs will vote on the deal at about 7pm. The two sides also agreed to continue working on technology that would do away with the need for border checks.
The move is aimed at addressing the concerns of hard-Brexiteers in her own Conservative Party and the Northern Irish Democratic Unionist Party (DUP), which provides her government with its majority in the House of Commons.