The Bradford City captain, Gary Jones, believes Arsenal's stars ought to be embarrassed by their Capital One Cup quarter-final exit at Valley Parade on Tuesday and he twisted the knife when he said that Torquay United had put up more of a fight against his team in Saturday's League Two meeting.

Bradford advanced on penalties after the tie had finished 1-1, with a clutch of Arsenal players underperforming on a bitterly cold night. It was the first time since the FA Cup defeat by Wrexham in 1992 that Arsenal had lost to opposition from English football's fourth tier and the result intensified the focus on Arsène Wenger and his methods.

The manager, though, retains the unwavering and unequivocal support of the club's majority shareholder, Stan Kroenke, and the chief executive, Ivan Gazidis. They have stood by him resolutely, including after the wobbles from earlier in the season, most notably the Premier League defeats at Norwich City and against Swansea City at home, and they will continue to do so. Gazidis did, however, apologise to Arsenal fans on Wednesday night for the defeat at Bradford, saying it was "not good enough".

Jones had played in Bradford's 1-0 home win over Torquay, which foreshadowed the Arsenal tie and the comparison that he drew was less than favourable for the Premier League club. "I think Torquay gave us a tougher game," Jones said. "It was like a role reversal against them. Torquay defended well against us and we defended really well against Arsenal. The lads were unbelievable."

Bradford's team on Tuesday cost £7,500 to assemble, as opposed to Arsenal's £65m – Wenger started with eight of his first-choice players – and Jones was asked whether Arsenal should be embarrassed.

"Without a shadow of a doubt," he replied. "No disrespect to us and our lads but they should be beating Bradford City. But team spirit means a lot and it can take a team a long way. We have a massive team spirit and we showed it.

"When we saw their team sheet, we couldn't believe it. Steve Parkin [the assistant manager] wrote their side down and at the bottom he wrote 'poor team', exclamation mark.

"I thought they would have a lot of youth players and a bit of experience but nothing like that turned up. They came here and it was probably the best chance of a trophy they'll have this year. I think it was Arsène Wenger's worst nightmare going to penalties. The lads were full of confidence and I think that showed."

Gazidis's apology came at a Christmas drink with Arsenal supporters. "I think I am frankly tired of getting up here and delivering the same message," the chief executive said. "Last night was not good enough and it made us all upset and angry. I would like to apologise to all of you, especially the fans who travelled up there. It was unbelievable support as ever and you deserved better.

"That is something we will work hard to put right. We all work here and are desperate to deliver the success and trophies we all want. Look into the eyes of the staff here, look at the way they talk. We will get this right."

Arsenal will seek to put the result behind them and focus on their league programme which, on paper at least, looks favourable. They play at Reading on Monday night and, afterwards, they face Wigan Athletic (away), West Ham United, Newcastle United (both home) and Southampton (away). Their inconsistency has been frustrating but the players, apart from a few grumbles, remain behind Wenger, who continued to stick up for them in public after Bradford.

There is, though, friction between Wenger and the assistant manager, Steve Bould, who was promoted in the summer from his role as the Under-18 academy coach. Wenger has, on occasion, betrayed his exasperation at questions that extol Bould's influence and he is clear about who is in charge of all technical aspects at the club. Bould replaced Wenger's long-standing assistant Pat Rice.

Wenger has also been vexed regularly since the Norwich defeat on 20 October about his team's inability to create chances and at Bradford they did not force a save from the goalkeeper, Matt Duke, until the 70th minute. "We were surprised how comfortable we were," Jones said."They didn't really have too many shots on goal. They tried to play too much football, especially down the middle. We defended really well and our organisation and discipline were really good.

"The only lapse of concentration was when the lad scored near the end. Apart from that I thought we defended really well."