The Italian added the world indoor title to his Olympic gold medal
The Italian won in 6.41 seconds – a new European record – and just three thousandths of a second ahead of defending champion Christian Coleman.
The USA’s Coleman – back after a ban for missed drugs tests – was beaten on the line while team-mate Marvin Bracy finished third in 6.44secs.
Jacobs was a surprise winner of the 100m at the Olympics last summer and believes victory in Belgrade on Saturday proves he is the best in the world.
“Of course, now I’m Olympic and world champion and this is so,” he told the BBC.
“After the two Olympic golds (100m and 4x100m relay), one in the World Indoors is amazing. I can’t wait to go to the outdoor and run and run. It’s a new European record, it’s amazing.
“They (the Americans) are great sprinters and athletes. It’s really good for me to compete with them.”
Coleman insisted he will improve ahead of the World Championships in Eugene in the summer and only needed to back himself following his ban, which saw him miss the Olympics.
“I feel I have something to prove to myself. I think really highly of myself and my talent speaks for itself. I want to be the best and keep working,” he told the BBC.
“I feel good, it was a season’s best and feel like I executed a good race and I’m still getting back in the swing of things.
“You want to come out and win, that is what everyone’s goal is. I felt we put on a good show and we’ll be there for the outdoors.”
Great Britain’s Adam Thomas, who qualified as fourth fastest, was fifth in 6.60s at the Stark Arena in Belgrade.
The 20-year-old, who won Olympic silver last year, was overwhelming favourite for the 800m title.
Hodgkinson warmed up at the Stark Arena but decided not to race in Belgrade following a recurrence of a quad injury picked up earlier in January.
She said: “I am in great shape so it is obviously a shame not to be competing here in Belgrade, but we decided to take this precautionary step in the best interests of the rest of the year ahead.
“I’m really pleased with my indoor season and it gives me a lot of confidence heading into the summer season which I am really looking forward to.
“Lastly, I would like to pass on my thanks to the medical team for their support and efforts to try and get me on the start line this week.”
She added to the BBC: “Luckily it’s not a major injury, it’s one to two weeks. I said to coach this morning if it was world outdoors ‘strap me up I’m running’. But I don’t want to have this problem to reoccur in the outdoor season.
“I made the right decision. I came here to get a gold medal, unfortunately I didn’t make the start line so it’s gutting.
“It was quite upsetting. I have never had to pull out of a race before. To come to a World Championships, my first Indoor World Championships as well, it’s not nice.
“My parents are here as well which is a little bit more upsetting but it’s about thinking about the bigger picture.”