The Russian side led through Alexander Ryazantsev's 29th-minute goal but faded late to lose 2-1 in the first leg of the last eight clash in Seville.
Carlos Bacca nodded in a 73rd-minute equaliser for the reigning champions before Denis Suarez's stunning volley settled the contest.
"The first half was very good for us," Villas-Boas told the club's official website.
"We played well, possessed the ball, were confident on defence. Of course, the fact that we were able to score also influenced the situation.
"There were chances to increase the lead, however we capitalised only on a single one of our chances.
"From the 60th minute, we started to have some problems. We looked worn out.
"We were forced to make some subs and this influenced the outcome of the match.
"Sevilla increased their control and had a lot of corners. It's pretty clear that our opponents gave out all their strength and that will make a difference in the return leg.
"The stadium wasn't the easiest to play in, and when our opponent scored their first goal, its emotions influenced both them and us as well."
Villas-Boas has called on Zenit's fans to produce an even better atmosphere in next week's second leg.
"When we return, we expect to have an incredible atmosphere and support from our fans," he said.
Villas-Boas also said that the leg injury suffered by Pavel Mogilevets looked "very serious" and that the club suspected he had torn a ligament.
On the result, Ryazantsev added: "Of course 1-1 or 1-0 would have been better.
"We have to score [in the second leg]. Sevilla has a really dangerous counter-attack, however I think that in St. Petersburg they are going to play a little bit differently.
"We definitely will too. We will have to attack and not forget about defence. We have to be careful."
