Former Labour MP Kelvin Hopkins who now sits as an independent also supported the government on one of the amendments.
Mrs May meanwhile revealed Donald Trump's advice to her on the Brexit negotiations had been to "sue the EU".
"Now, if they do whatever they do, they have to..."
Tory policy, just like Labour policy, is to leave the Single Market and the Customs Union, and we have a duty to work with whoever is willing to work with us, to make sure that doesn't happen, for the sake of our economy.
But the minister added: "I don't know where we go from here". The Chequers agreement, named after the prime minister's countryside retreat, shifted the government's position in a soft Brexit direction.
The prime minister insisted she was confident Britain had enough time to negotiate a deal with the European Union before leaving in March next year.
But despite speculation suggesting a potential setback for Mrs May, the trade bill amendment was defeated by 307 votes to 301.
Mr Edwards said there was no point in holding another general election as Labour "shares the exact same Brexit policy as the Tories". "We'll have an idea of the numbers, I suppose, at 10 o'clock on Monday evening".
The amendment would have required the government to try to negotiate a customs union arrangement with the EU if, by January 21, 2019, it had failed to negotiate a frictionless free trade deal with the bloc.
Parliament voted 307 to 301 against an amendment to trade legislation in a vote that has been closely followed by currency markets, and was seen as a test of May's ability to deliver a smooth British exit from the European Union and keep her job.
Mrs May still hopes to present the final Brexit deal to parliament in the autumn.
On Monday night, Mrs May bowed to pressure and accepted four amendments from Brexiteers to a key piece of legislation, including one aiming to block her plan to lock the United Kingdom in a customs deal with the European Union (EU).
But having trusted the Prime Minister to deliver on her "Brexit means Brexit" promise, he now finds her latest proposals unacceptable.
Boris Johnson is to make a personal statement to the House of Commons on his resignation as foreign secretary as Theresa May faces a grilling by senior MPs. "Whether that's this autumn or next spring, I can't say".
Most traders believe Britain will avoid a so-called hard exit from the European Union, in which trade ties are minimal, but the pound remains highly sensitive to political uncertainty and bearish sentiment. "It's time we shared their confidence".
Ahead of a crucial week in Parliament, the Prime Minister acknowledged feelings in the party were running high, but said her plan offered a "hard-headed and practical" way forward.
Parliament could go into recess - a week earlier than planned.
But the suggestion that MPs take extra time off during one of the most critical - and busiest - periods in recent parliamentary history has gone down badly even among members of her own party.
"My job as Prime Minister is to deliver for them, but also I've got to be hard-headed and practical about this and do it in a way that ensures we get the best interests for the United Kingdom".