There was an interview right after Spider-Man: Homecoming, where Kevin Feige confirmed that this Spidey does have the spider-sense.
I recently asked Marvel Studios president and Homecoming producer
Kevin Feige if the Spider-sense exists in the MCU or if the need for
such a super-power was negated thanks to Spidey's costume now being
decked out with AI and other Stark tech that aids him in exploring his
"No, I think he has it. And I think he has it with or without that
suit," Feige told IGN. "I think how we explore it in a cinematic sense
will change. I mean, that was sort of a big showy part of previous
versions and we thought that we'd make it more of an internal, sort of
second nature thing for him."
Feige continued, "But there are ways coming up that will slowly hint
at that and also just make it part of his, you know, his natural
abilities. But we don't know that if it will be -- I think we'll
explore it further down the line, but [it's] definitely him not the
If that is the case, we can safely conclude that this version of Spider-Man does indeed have the spider-sense.
So by extension, his "reflexes" in Civil War would certainly be an outcome of the spider-sense.
So far that I was able to search, I did not find any commentary specifically on that incidence in Civil War.