It was a reference to Webb awarding a penalty to Manchester United against Liverpool and sending off Reds captain Steven Gerrard in an FA Cup tie in January.
Wilshere's team-mate Abou Diaby was sent off in Saturday's pulsating game at St James' Park where Newcastle were also awarded two penalties.
The French midfielder got his marching orders for reacting to a strong tackle from Newcastle's Joey Barton.
Arsenal boss Arsene Wenger said Barton was lucky to stay on the pitch, but the Newcastle player pointed to Arsenal's lack of discipline.
The Magpies trailed 4-0 at the break but Diaby was dismissed five minutes after the restart and the Gunners capitulated as the home team mounted a stunning fightback.
"[At half-time] I was thinking that I hope it does not get silly, double figures, but football is a funny game," said Barton.
Newcastle just trying to avoid cricket score - Barton
"[There was] a lack of discipline in their side, we got a goal and then the Geordie nation got behind us.
"Diaby had taken a bad touch and I had to try to win the ball. I don't understand the guy's reaction. He will probably cop a couple of weeks' fine because it has probably cost them."
Newcastle manager Alan Pardew argued that Barton was entitled to challenge for the ball.
"You have got to make a challenge. It was a proper, honest challenge - I have seen it two or three times and there's nothing wrong with it," said Pardew.
"He goes for the ball, he wins the ball clean. He is aggressive in the challenge, he takes a little bit of the player after he has won the ball."
But Wenger had a different interpretation, hinting that Barton was guilty of a dangerous tackle.
"It was a completely unnecessary sending-off, and I believe Barton was very lucky to stay on the pitch for the tackle on Diaby," the Frenchman stated.
"It's a shame [for Diaby] because he had a great first half. Sometimes you could understand when he has a bad game but he had an outstanding game.
"I explain that by the fact that he has been injured so many times from bad tackles that he lost a bit quickly his nerves because this boy has been out for a long time and many times."
It's one of the most remarkable games I have ever been involved in in my life
Newcastle boss Alan Pardew
Wilshere also took aim at referee Dowd for not sending off Newcastle skipper Kevin Nolan for an incident that followed the first of Barton's two penalties.
Barton and Nolan both ran towards the Arsenal goal with the aim of collecting the ball and ensuring a quick restart.
But Arsenal keeper Wojciech Szczesny held the ball and refused to hand it over, prompting Nolan to try to wrestle it off him.
The Newcastle player appeared to have his hands around his opponent's neck as he did so and Dowd booked both players after the incident.
Although the Twitter posts were later taken down, Wilshere added: "If Diaby is sent off, what's the difference between that and Kevin Nolan's challenge on our keeper??#joke."
Despite losing a four-goal lead, there was consolation for Arsenal later in the day when Manchester United lost 2-1 at Wolves - leaving the Gunners only four points behind the leaders and
Wilshere to surmise in a later tweet
"very strange day. We are gutted like all of the AFC fans, but on the positive side we have a point and man u lost! Lets learn and move forward!".
Wilshere's posts came after an incredible match in which Theo Walcott opened the scoring for Arsenal after 44 seconds at Newcastle and the Gunners raced into a 3-0 lead in 10 minutes.
Robin van Persie's second goal of the game ensured the visitors took a 4-0 lead into the break, but they were powerless to prevent Newcastle scoring four times in 19 second-half minutes.
Arsenal 'panic' frustrates Wenger
"It's one of the most remarkable games I have ever been involved in in my life - and I am talking about Sunday mornings as well, 7-7s, I have had, everything," Pardew commented.
Wenger conceded that his team lost their composure after the break and said he did not know how costly his team's failure to win the game would prove to be to their Premier League ambitions.
"Mathematically, two points; psychologically, the damage is a bit bigger because everybody is, of course, very, very disappointed in the dressing room," he said.
"Football is unpredictable sometimes and I felt that it became difficult at 4-1.
"Not at 4-0, we were quite comfortable. After, we panicked a little bit. I feel we were a little bit unlucky as well on some occasions but we have to take it on the chin and keep the positives from the game.
"It was a good game with a lot of intensity. There were many turning points, I feel of course the sending off was detrimental to us and the first penalty as well gave Newcastle hope."
Wenger refused to comment further on the two penalties awarded to Newcastle. "My opinion is not important," he added.
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