"The problem is not Wembley, it is us," he said, disturbed at his side's lack of concentration. "We must take responsibility, it is our fault. You will pay if you are not focused for 90 or 95 minutes. When you have three points in your hand, it is difficult to accept."
Kieran Trippier, the former Burnley right back, was culpable in the closing stages, as Spurs left too much space for their opponents to attack. Brady had already opened up Tottenham with a pass that put Wood through on goal, only for Lloris to fly out of the penalty area and make a crunching but legitimate tackle.
Moments later, Trippier failed to track Wood as he ran on to another Brady pass to shoot past the French keeper.
It left Pochettino frustrated and disappointed in equal measure, although he made the point that three games into the season is far too early to predict how things will finish next May, citing Chelsea's slow start at this stage last year before becoming champions.
"It is still early to analyse how things will be," he said. "We need to work hard and be more strong. Because that is a mental problem that we need to fix."
He expects to make more signings before Thursday's transfer deadline, having given a brief debut to Davinson Sanchez, the Colombian defender signed from Ajax last week.
It was the sort of game that Spurs closed out successfully in the past, and a team challenging for the title needs to do. They certainly had enough chances to do so once Alli put them ahead in the 49th minute with his second goal of the season.
Ben Mee failed to deal with a corner from Christian Eriksen and the ball fell to Alli, whose first close-range effort was blocked by Matthew Lowton in front of goal. Alli was given a second chance and made no mistake, smashing the ball home.
That should have been the springboard for more. Harry Kane, yet to score a Premier League goal in August after three years of trying, curled a shot just wide and then had a powerful first-time shot saved brilliantly by Tom Heaton, diving low to his right.
The keeper then made an even better save to tip over an angled drive from Eriksen, and spread himself to keep out Kane when he had a free run on goal. Alli had another shot saved, Kane put a free-kick into the defensive wall, and Burnley sensed they might get something from the game.
Lloris made a spectacular stop to keep out a stinging shot from Brady, who got his revenge in stoppage time when he set up Wood for an equaliser that Burnley boss Dyche believed his side deserved.
"Tottenham controlled the first half, as you might expect, but we came in at half-time intact, which was our plan. As the game went on, we kept going for it and I was pleased Woody scored. On balance, I think we deserved it."
Burnley had upset the form book already by winning at Chelsea and now they have upset Tottenham at their temporary home - though Dyche agreed with Pochettino that Wembley should not be a factor.
Whether Tottenham's supporters will see it the same way is another matter.
