The overwhelming favourites ahead of the fixture, Spurs looked to have booked a ticket for the quarter finals as they led 2-0 at half-time, courtesy of goals from Moussa Sissoko and Dele Alli.
However, a quickfire double from Andre Ayew restored parity halfway through the second half, before Angelo Ogbonna headed in the winner, sparking jubilant scenes among the travelling supporters and relieving the pressure on Slaven Bilic, who had arrived at Wembley with his job on the line following a difficult start to the season.
The comeback handed the Hammers their first win in five games in all competitions and Redknapp claimed Spurs' demise smacked of arrogance.
"I'd go as far as to there was too much arrogance from Tottenham," he told Sky Sports when asked if his former side had paid for their complacency.
"They got all the plaudits, and rightfully so, after Sunday - they were fantastic and outworked Liverpool in every department. But today's performance was a shocker. To go from being two goals up and completely in control to losing the game showed signed of arrogance and complacency."
Ahead of the game, Spurs manager Mauricio Pochettino challenged his side to win one of the "two real trophies" in their sights this season, dismissing the importance of the two domestic cups.
When it was put to Pochettino that winning the FA Cup or the Carabao Cup could prove to be a springboard for his side, he quickly dismissed the notion, suggesting only winning the Premier League or the Champions League would "really change the life" of the club.
While Redknapp agreed with the Argentine in prioritising the Premier League and the Champions League, he insisted Spurs had missed a great opportunity to win a first trophy under Pochettino, which would also been the club's first trophy in a decade.
"You could say that this competition is the last of Pochettino's priorities but it's not, this is a great marker to put down", he explained.
"With a trophy in the bank, it could have provided a feel-good factor for the rest of the season to go on and win more."
Redknapp, who played for Spurs between 2002 and 2005, added failure to win either League Cup or the FA Cup would also pile further pressure on Pochettino's team and the former Southampton manager, who is yet to win a major honour as manager, could soon face questions over his tenure.
"With the players he has got, it won't be long until he's judged on what he's won," he said. "If you don't win things with the players you've got it won't be long until you come in for criticism."
Last Sunday (22 October), Spurs moved up to second in the Premier League table alongside Manchester United after demolishing Liverpool 4-1 and will travel to Old Trafford this coming Saturday, a ground where they have not scored in the last three league visits.
The duo arrive into the fixtures five points behind Manchester City but Spurs could have to do without Harry Kane, after Pochettino admitted his talismanic striker was struggling with an injury.
"We need to assess the team because we were forced to rotate for different reasons and we'll see what happens in the next few days," he said. Kane has been in fine form for Tottenham this season, scoring 13 goals in 12 appearances in all competitions this season. This also includes five goals in the Champions League from three starts.