Sammie Szmodics and Liam Delap gave Ipswich a 2-0 lead at Tottenham Hotspur Stadium as Spurs fell short, despite Rodrigo Bentancur returning a goal in the second half.
It was Tottenham's third defeat in its past five matches across all competitions and leaves Postecoglou's side in 10th place in the Premier League with 16 points from 11 matches.
"It's hugely disappointing. We started slow and passively. We gave ourselves a mountain to climb. In the second half, we had chances but didn’t do enough to win the game," the Australian said.
"It's just down to me, I’m not getting consistent performances from the players. It’s something I need to address. I’m the person in charge, so that’s usually the way it goes. I take responsibility.
"It’s very disappointing. Before they scored, we had a couple of chances. We conceded a very sloppy goal that shouldn’t happen, it’s more than disappointing," the Korean said.
It was the fifth straight Premier League home game in which Tottenham had conceded first and proved costly this time around.
Szmodics opened the scoring for the visitors with an acrobatic finish just past the half-hour mark before Delap doubled Ipswich's advantage prior to the interval.
Spurs, who saw a Dominic Solanke goal overturned by VAR for handball early in the second half, handed a lifeline when Rodrigo Bentancur headed in from a corner to cut the deficit in the 69th minute.
Solanke had a great chance to restore parity in stoppage-time, but Arijanet Muric made a fantastic stop as Ipswich held on for a memorable triumph.
"I am so proud. It's been a long time for the club, 22 years since the supporters saw a win in the Premier League. The journey to get to here has been a big one," McKenna said after the game.
"It feels a bit special today. We believe we’ve been improving. We had picked up five out of 10 results but were waiting for the first win to validate the work and improvements. It’s a massive moment, one to cherish.
"The first half, it was everything [you'd want] for a team in our context against Tottenham. There was a good balance. There were moments where we played a good tempo and moments where we slowed it down.
"It was a top half away from home. In the second half we showed good resilience. We didn’t give away too many chances. We defended well and were a threat on the break. A top performance.
"It was really important [to hang on]. If we’d conceded at the end it would have been a big blow. To get one over the line like that we’ve deserved for quite a few weeks. A big boost for everyone.