Robertson, 26, took gold in 6.57 seconds, with Theo Etienne second and 38-year-old Dwain Chambers third.
James Dasaolu, 2016's winner, and CJ Ujah did not attend the event at Sheffield's English Institute of Sport.
Asha Philip comfortably retained the women's 60m crown in 7.19 seconds.
Kilty, who took the world 60m title in Prague in 2015, ran the fastest time in the heats, but again bowed out before the final due to a clear false start.
However, he will not withdraw from next month's European Indoor Championships as he did from the World Indoors after last year's disqualification, and is hopeful of earning selection.
"If the selectors can have faith, I'm not going to make a decision like I did last year to withdraw from the championships," he said.
"This time I stupidly, more than trying to win the race, was chasing a really fast time.
"I've never won a national championships, I've won everything else and I just don't understand - honestly I'm just baffled. I've just been stupid."
However, Robertson told BBC Radio 5 live that Kilty's disqualification had no bearing on his victory.
"I was very, very confident of the win, even if Kilty was in it I knew I could win," he said.
Philip - who beat Dina Asher-Smith to claim her first title last year - was not faced with her rival this time round, and raced clear of the field to win with ease.
The victory has also secured her automatic selection for March's European Indoors in Belgrade.
"I'm over the moon, I'm excited, I didn't get to go [to the European Indoor Championships] two years ago, I want to go this year," she told 5 live.
Andrew Pozzi ran 7.44 seconds to equal the fastest indoor 60m hurdles of the year and remain British champion.
Eilish McColgan pipped her Scottish compatriot Steph Twell to the women's 3,000m title on the final straight, while Morgan Lake and Beth Partridge decided to share the high jump title after it went to a jump-off.
The 200m and 400m races will be held on Sunday, as well as the men's 800m and 3,000m finals, streamed live via BBC Sport on connected TV and online from 11:55 GMT.
Andrew Robertson won his first British 60m title at the indoor team trials, as 2015 world champion Richard Kilty was disqualified in the final for a false start for the second year in a row.