Gibson, from Prestwick, said he was "overwhelmed" after the first part of the competition, which will conclude tomorrow evening with the free dance.
He said: "We just can't believe it, we are really, really, happy. We beat our personal best by five points - and at the Europeans too. It's amazing."
Gibson earlier this week told The Scotsman how the couple had made changes to the music and choreography of the first part of their rhythm dance programme ahead of the championships in a bid to create more impact with the judges.
He said: "We're just delighted that this paid off. It went down well with the crowd too, we are just overwhelmed."
The couple did not compete in the European Championships last year, but placed fifteenth in the 2016/17 season in the competition.
Fear added: "We would be really happy with a top ten finish. If we did that, it would mean there would automatically be two places for British couples next season too."
She added: "Usually, the rhythm dance is the difficult one and doesn't go down as well with the crowd, but today it did and we are now just looking forward to skating our free programme. We have loved skating to this music"
Teammates Robynne Tweedale and Joseph Buckland, from London, placed 18th in the rhythm dance with a score of 57.38.
Silver Olympic medallist Gabrielle Papadakis and Guillaume Cizeron from France are standing in first place after the rhythm dance, scoring 84.79, followed by Alexandra Stepanova and Ivan Bukin from Russia.
Gibson and Fear, who are trained in Montreal by Romain Haguenauerf, skated their rhythm dance to music by the Gypsy Kings. They previously had a personal best in this section of the competition of 64.71, achieved at Grand Prix competition Skate America earlier this season. Their free dance, skated to a medly of music by Donna Summers, has previously gone down very well with both judges and the crowd.
British skaters Zoe Jones - the championship's oldest competitor at 39 - and partner Christopher Boyardji gave an emotional performance in the free skate of the pairs competition, but placed 10th out of ten teams to compete after scoring 138.85. The eleventh competitors, Miriam Ziegler and Severin Kiefer, withdrew citing "emotional distress".
Meanwhile, Irish men's champion, Conor Stakelum, who trains at Ice Dundee with Debi and Simon Briggs, did not qualify for the free skate after placing 33rd. Graham Newberry who skates for Britain, did qualify with a short programme placing of 22nd, and will take part in the free skate on Saturday.
The free skate in the ice dance competition will take place tomorrow night at Minsk Ice Arena.
Dundee skater Natasha McKay is due to compete in the free skate programme of the ladies's singles competition tonight.