The Posh went into the second leg of their semi-final tie with a 4-0 lead over Darren Moore’s men.
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But a remarkable turnaround at Hillsborough saw Sheffield Wednesday send the game to extra-time thanks to a 98th-minute Liam Palmer strike.
It then went to penalties after finishing 5-1 at the end of 120 minutes.
And Wednesday clinched a 5-3 victory in the shoot-out to book their place at Wembley where they will face Barnsley or Bolton.
Carragher tweeted his excitement when Palmer’s stoppage-time goal brought the tie level, taking to Twitter to simply say: “OMG!!!!!!!”
He then followed that up moments later with a cheeky dig, adding: “Fergie time!”
The post was in reference to the dad of the Peterborough boss, Sir Alex Ferguson.
The ex-Manchester United manager was famous for clinching games in added time during his 27 years at Old Trafford.
It was such a regular occurrence that the phrase “Fergie time” was coined.
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But on this occasion that very same “Fergie time” came back to bite son Darren as Peterborough suffered play-off heartbreak.
They were hoping to return to the Championship at the first time of asking following relegation from the second tier last season.
But despite sneaking into the play-offs on the final day, they came up short against Wednesday in an all-time classic tie.
Ferguson returned to the club for a FOURTH spell in January, 11 months after his third stint came to an abrupt end.
