Spurs had defender Davinson Sanchez dismissed for elbowing Richarlison in the 52nd minute, but battled to secure an away point and move up to sixth in the Premier League.
The hosts had taken the lead in the 13th minute when Christian Kabasele headed in from six yards from Tom Cleverley’s left-wing corner.
Watford thought they should have had a penalty when Richarlison appeared to be fouled, but it was not given, and less than a minute later Tottenham’s Son Heung-min finished off Christian Eriksen’s cross to equalise.
After Sanchez’s red card, Abdoulaye Doucoure hit the post for the hosts with a 25-yard strike but Watford could not find a winning goal.
“We’re in a period now that we are suffering things against us,” said Pochettino. “It is good for the group to try and learn. It is a long-term project. It will be good for the future. You want to win but in tough moments you learn.
“I think it wasn’t intentional for Davinson to do what he did. I am not going to talk about the refereeing decision any more.
“For me one of the first questions is about a controversial situation. My decision is not going to change it.”
This was Tottenham’s fourth Premier League match without a victory, although they do climb above Burnley, who lost 1-0 at Leicester, while Watford remain eighth.