Abbott, inside centre Ngani Laumape and flyhalf Beauden Barrett crossed in the first half for the visitors, while fullback Jordie Barrett slotted four conversions.
Scrumhalf Augustine Pulu, wing Melani Nanai and lock Scott Scrafton all scored tries for the home side, with Piers Francis adding two conversions while replacement flyhalf Bryn Gatland added a penalty and conversion.
“We knew it was going to be an 80-minute performance for us,” Hurricanes captain TJ Perenara said. “They really took us to the end, but I’m proud of the boys and the way they fought.
“We were under the pump for a big part of the second half, but we were able to do enough to get over the line.”
The Hurricanes, who had been sublime in the first half against the Waratahs last week, looked muddled on Saturday, had their potent running game shut down and lived off the Blues’ errors.
While Hurricanes coach Chris Boyd might be pleased with the win, he will be concerned around his flyhalf’s disciplinary record after Barrett received a first half yellow card for a deliberate knockdown while on defence.
The 25-year-old was sent off last week against the Waratahs after he received two yellow cards for the same offence.
A disciplinary hearing earlier this week decided not to impose any other sanction, though they may take a dimmer outlook should he appear before them again.
Barrett, however, had demonstrated how important he was to the Hurricanes attack, setting up first-half tries for Abbott and Laumape, with his pace getting him behind the Blues defence.
He then crossed for his own try when Sonny Bill Williams dropped the ball in midfield and the flyhalf pounced to run 45 metres untouched.
While Pulu and Nanai finished off two of the Blues’ try-scoring chances in the first half, the home side blew several more opportunities that could have put them in control of the game.
The wastefulness of both sides continued in the second half, with the only points for the first 25 minutes a penalty to debutant Gatland, the son of British and Irish Lions coach Warren.
Scrafton then barrelled over to give the Blues a 24-22 lead with 15 minutes remaining, before Barrett again exploited some space behind the line to launch another counter-attack that sparked 16 phases of play and ended with Abbott crossing in the corner.
The Blues remain at the bottom of the