A defiant Jeremy Corbyn is to tell critics in his shadow cabinet that his leadership has been strengthened by Labour’s emphatic victory in the Oldham byelection and by his success in winning a majority in the party for his opposition to airstrikes.

The party leadership said the win in Oldham West and Royton in the early hours of Friday morning had proved that Labour is electable, while shadow cabinet sceptics acknowledged that Corbyn would remain in place for “far longer” than they had expected.

“Jim McMahon’s victory makes a nonsense of the claim that Jeremy Corbyn’s Labour party is unelectable,” a spokesperson for Corbyn said. “Now Labour has increased its share of the vote in Oldham, Jeremy Corbyn’s leadership is stronger as a result.”

Ken Livingstone, the former London mayor who is a close ally of Corbyn’s, said that the win would change the dynamics in the party. In a Guardian interview he said: “All those MPs who genuinely believe that we can’t win with Jeremy, and that’s fuelled with opposition, might start to rethink. If it starts to look as if Jeremy can win the next election, they will all want jobs – they’ll get in line.”

The leadership moved quickly to exploit Labour’s win in Oldham in which McMahon, the leader of Oldham council, won a 10,722 majority in Thursday’s byelection, comfortably seeing off a challenge from Ukip.

McMahon increased Labour’s share of the vote from the 54.8% won by the late Michael Meacher in the general election to 62.1%. McMahon secured fewer votes than Meacher (17,209 compared with 23,630) because the turnout was lower: 40.26%, down from 59.6% in May. It was, however, a high turnout for a byelection.

Corbyn, who made a brief visit to the constituency on Friday morning to congratulate McMahon, said the decisive victory is proof that his party is a broad church with deep-rooted support across the country.

Flanked by placard-waving supporters on the steps of Chadderton town hall, Corbyn said: “This campaign shows just how strong our party is, not just here in Oldham, but all over the country. We’ve driven the Tories back on tax credits, on police cuts, on their whole austerity agenda and narrative. It shows just how strong, how deep-rooted and how broad our party, the Labour party is, for the whole of Britain.”

The leadership said that the byelection victory capped a difficult, but ultimately successful, week for Corbyn in which secured the support of a majority of the party in its three key sections for his stance against the RAF airstrikes in Syria.

He won over a majority of the shadow cabinet, whose members voted by 17 to 11 against the strikes; he won support among two thirds of Labour MPs, with 66 voting for airstrikes against Isis in Syria; and opinion polls have shown a decisive majority of party members oppose the airstrikes.

The spokesperson for Corbyn said: “Jeremy Corbyn won a large majority this week for his stand against bombing in Syria in every part of the Labour party, inside and outside parliament, including the shadow cabinet. Jeremy Corbyn led the campaign against tax credits and police cuts which forced the government to U-turn.”

The Labour leader is likely to assert his authority over his opponents in the shadow cabinet by making clear that Hilary Benn, the shadow foreign secretary, will need to hold the government to account on Syria in line with the thinking of the Labour party.

Corbyn will be expecting that Benn will focus, after his barnstorming Commons speech, on pressing the government in four areas in the party’s “common position”. These are on the deployment of regional ground forces; the need for a comprehensive negotiated settlement in Syria; on the need for the military action to be subordinate to the diplomatic and political process; and the need to focus on the refugee crisis.

The message from the leadership came after Corbyn’s critics, who had predicted a narrow win for Labour in the byelection, were confounded by the scale of McMahon’s victory. Shadow cabinet ministers had predicted a collapse in the white working class vote, with many of those voters turning to Ukip. They said Labour would inch across the line on the basis of black and minority ethnic votes.

But McMahon’s 10,722 majority showed that the party’s vote held up and actually increased in percentage terms. Livingstone mocked Corbyn’s critics: “We were being told the white working class didn’t like Jeremy!”

One member of the shadow cabinet said: “You can’t extrapolate from one byelection. However, Jeremy was on our leaflets but more importantly he was on every Ukip leaflet that went through every door. You’d have to be living on Mars to not be aware that Jeremy Corbyn is leader of the Labour party. So people didn’t vote in ignorance.”

Oldham West and Royton Ukip candidate John Bickley at the byelection results with Sir Oink Alot of the Official Monster Raving Party (left) and the Liberal Democrats’ Jane Brophy. Photograph: Gareth Fuller/PA

In a sign of the continuing bitterness in the party, one critic of Corbyn said there was some merit in the claim by Farage that the result lacked legitimacy because of the high number of postal ballots from black and minority ethnic voters. The Ukip leader, whose candidate, John Bickley, finished a distant second on 6,487, is to lodge a formal complaint.

The Labour critic said that BME voters were “massively engaged” to vote because the former BNP leader Nick Griffin received 16% of the vote when he stood in the constituency in 2001 after the Oldham race riots. The Labour figure said that sampling in the Werneth ward, whose population is almost entirely from BME communities, found that of the 814 voters, 812 voted Labour. By contrast there was “real resistance to Labour” in white working class areas where voters abstained rather than voting for Ukip.