Goals from Watkins and Morgan Rogers, the latter in the first-minute of the second half, appeared to have Villa on course for a comfortable win to boost their Champions League aspirations.
But the visiting Bees then struck three times in nine minutes through Zanka, Bryan Mbeumo and Yoane Wissa to turn the match on its head.
Watkins then equalised for Villa with his 18th league goal of the season but Unai Emery's team were unable to find a winner.
It was just the second time this season they had dropped points from a winning position and Watkins said: "It's frustrating. We lacked that big team mentality where they kill games off, which is really disappointing.
"I'm not belittling my team. I'm part of it. I feel we need to somehow figure out when we're 2-0 up how to just shut up shop.
"I felt when it went to 3-3 that we would go on and win. We had a lot of the ball and we were creating chances."
The draw saw fourth-placed Villa open their advantage over Tottenham to three points but their rivals, who play Nottingham Forest on Sunday, have two matches in hand.
Watkins said: "There is a lot of football to be played. We're battling with Tottenham who have got a great squad. It's going to be tight. We need to win the next six games.
"Two-nil is always a dangerous scoreline. We came out in the second half and scored and then we seemed to relax as a team.
"I don't know if it's the scoreline or what but we always come out slow. I said to the boys at the start of the second half we need to start quick.
"It's poor. It was too easy for them to get in and score. It's really disappointing because it could have been a really massive three points. We've been 2-0 up against Manchester United away and conceded silly goals. We have done it here today.
"We've got a point but it should have been three. These are the games we need to be winning. That's what the top team do. They finish it off, kill it off. They are smart. Maybe we just don't have that maturity yet or game intelligence."
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