Fourteen Labour MPs voted with Theresa May’s government to prevent parliament from taking steps to prevent a no-deal Brexit by extending the article 50 negotiating period in order to agree an alternative deal with the EU.

The former minister Caroline Flint and Laura Smith, the MP for Crewe and Nantwich, joined with the veteran Eurosceptics Kate Hoey, Graham Stringer and Dennis Skinner in voting down Yvette Cooper’s amendment by 321 votes to 298.

Their votes helped cancel out those of 17 Conservative rebels who voted for Cooper’s amendment, which Labour’s frontbench had decided it would support earlier in the day to “reduce the threat of the chaos of a no deal”.

A sizeable Labour rebellion had been expected for several days, despite the efforts of Cooper to calm concerned MPs in her party by telling the Commons it only gave parliament the right to vote on whether to extend article 50 if time ran out.

But she failed to persuade the doubters who defied the party whip, and whose numbers were noticeably larger than the half dozen or so Labour MPs who previously voted with the government before on key Brexit votes.

Reacting to the result, Jeremy Corbyn chose to focus on the success of another no-deal amendment, in the name of the Conservative backbencher Dame Caroline Spelman, which was carried by 318 votes to 310.

The Labour leader said he would meet May, having refused to do so for a fortnight, because “parliament has voted to remove the immediate threat of crashing out without a deal on 29 March”.

The Spelman amendment, which was also supported by Labour’s Jack Dromey, invited MPs to reject the UK leaving the EU without agreeing a deal, but is not legally binding on the government. The Cooper amendment was intended to create parliamentary time for MPs to legislate to prevent no deal.

Other Labour MPs who voted with the government to vote down the Cooper amendment included Ian Austin, Sir Kevin Barron, Ronnie Campbell, Rosie Cooper, Jim Fitzpatrick, Roger Godsiff, Stephen Hepburn, John Mann and Gareth Snell.

After the vote, Cooper said: “[The] Commons has voted tonight against no deal, though there was not the support this evening for a legislative safeguard. I’m still very worried that [the] PM does not have a workable plan to prevent a cliff edge at the end of March.”

Seven Labour MPs – all well established rebels against the party whip on Brexit – also voted in favour of Sir Graham Brady’s amendment, which called on the government to find alternative arrangements to replace the Irish backstop. They were Austin, Barron, Fitzpatrick, Godsiff, Hoey, Mann and Stringer.

Downing Street is expected to concentrate attention on the group of Labour MPs, because they may be needed to help May get any final Brexit deal approved, if she can woo them with extra undertakings on workers rights.

May repeatedly signalled her intentions in the Commons debate, telling the Labour MP Melanie Onn: “We are looking at ways in which we can give that assurance in relation to workers’ rights.”

Downing Street still wants the support of Labour members because few believe it will be possible to eliminate the Conservative rebels, even if May were to succeed in removing the Irish backstop in last-ditch negotiations with the EU.

Labour had tried to ease the concerns of those worried about the Cooper amendment by saying it wanted to shorten the nine-month extension to the article 50 negotiating period to three months to avoid accusations it wanted to permanently delay Brexit.

The shadow chancellor, John McDonnell, said Labour wanted ideally to reduce the extension period to “something around” three months because the party was anxious “not to be seen in any way to be kicking the can down the road”.

The senior Labour MP said while the party wanted to be “flexible about the timescale in which we can achieve an alternative [Brexit deal]”, it wanted an extension to be as short as possible.

Labour’s shadow cabinet met on Tuesday morning to discuss tactics before the Brexit debate in which MPs voted on a range of amendments selected by the Speaker at lunchtime. A party source said the Cooper bill “could give MPs a temporary window to agree a deal that can bring the country together”.

The backing of Labour’s frontbench for Cooper’s amendment had not been certain after Jon Trickett, the shadow Cabinet Office minster, said on Monday voters in his constituency would regard support for the measure as a failure to respect the result of the 2016 referendum.

Trickett said he felt “it may look to people as if we’re trying to somehow remove the earlier decision, which was to Brexit”, by voting for an amendment intended to extend article 50.