Lucas Paqueta struck from the penalty spot on 87 minutes, but Enes Unal found an equaliser for Bournemouth three minutes later.
Lopetegui later said, "We were very close to winning against a very good team. That is why you are very close there is frustration to lose these two points. But there are positives against a very demanding team in Bournemouth.
"We deserved to score. In the second half we knew they were going to push more and they did. I prefer to be positive and look forward.
"We deserved to win at Leicester. Today we did well too. Above all I want to highlight the behaviour of the group. All the players want to work very hard to do their best for the team. This is the first step to be strong and a better team. We are going to keep this positive mentality."