In an interview with Sky News earlier, Chancellor Philip Hammond said the analysis showed the prime minister's Brexit deal offers the political benefits of being outside the European Union with "very little economic cost".
With some Remainers also failing to vote, and five Tory MPs voting with the opposition, the Government fell to a 303 to 258 vote defeat.
Parliament "has passed legislation that we are leaving on 29 March and it put that date on the face of the bill by a large margin", Reuters quoted Barclay as saying in parliament.
"She can not keep on just running down the clock and hoping that something will turn up that will save her day and save her face".
With the prime minister estimated to be more than 60 votes short of the majority she needs for the House of Commons to approve her Brexit deal, Labour shadow chancellor John McDonnell suggested a new referendum might be "inevitable".
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While the Spelman amendment was technically not legally binding, it became a sticking point for the rightwing European Research Group (ERG), a cadre of staunchly pro-Brexit Conservatives; who abstained on the principal motion.
He said May believed that her party still supported her bid to continue negotiating in Brussels, but that they had rejected the motion because they were concerned about taking a "no deal" Brexit off the table at this stage.
Many back the fact it reiterates support for the direction agreed on January 29, when MPs supported an amendment authorising Mrs May to return to Brussels to renegotiate the controversial Irish backstop.
"However, [this] vote has proved that there is no majority for the Prime Minister's course of action". "The legislation, frankly, takes precedence over that motion".
"We continue to think it likely that, rather than allowing the vote and consequent ministerial resignations to occur, PM May will attempt to forestall by stating that she will seek an extension herself".
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An amendment tabled by the SNP which would delay the implementation of Article 50 by at least three months.
"Theresa May's own deal is opposed by people on all sides of this debate and her only tactic is to run the clock down in the hope of forcing the country to agree to it".
In an interview with Parliament's The House magazine, conducted before the vote, he said they had celebrated after Mrs May was defeated in last month's vote and were not true Conservatives.
The Liberal Democrats' Tom Brake asked: "In what way can we as members of Parliament, who have already voted to rule out no deal, can ensure that the government listen to that and respond appropriately?"
Theresa May has herself changed tact to warn of the threat to the union if the United Kingdom were to leave the EU without a deal.
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The Government has not even published any economic analysis of Theresa May's Brexit deal.