Blair’s team squeezed home 29-26, securing a precious bonus point in the process, but they trailed for spells in each half against the Italian side who also drew level with six minutes of the match remaining.
Edinburgh needed a 76th-minute penalty from Jaco van der Walt to win as Zebre made a mockery of their lowly standing as the only winless team in the United Rugby Championship.
Edinburgh got off to the perfect start with Mark Bennett’s try in the first minute but looked laboured thereafter.
“We scored early and there was an air of over confidence during the week coming into Zebre game,” said Blair, the Edinburgh head coach.
“Eyes were maybe on the Ulster and Glasgow games coming up. Scoring early sometimes everybody relaxes a little bit.
“We were off it today in terms of our speed to contact and it was not great but it could be a lot worse. We still walk away with a bonus point victory and we are in the top eight and still have the potential to go top four so all is not lost but we have to get better.”
The result lifts Edinburgh into sixth place in the URC standings, with seventh-placed Sharks due to host leaders Leinster on Saturday. The top eight sides qualify for the play-offs, with the top four being granted home advantage.
Edinburgh, whose final two regular season games are at home to fourth-placed Ulster and fifth-placed Glasgow Warriors, played the final 36 minutes with 14 men following Marshall Sykes’ red card for a dangerous clearout in which his shoulder made contact with Danilo Fischetti’s head.
“We kept making the same mistakes but we cannot keep doing that but at the same time I thought there was still some very good game management when we were down to 14 men,” added Blair.
Edinburgh, who also had Glen Young yellow-carded in the first half for a high tackle which saw Zebre awarded a penalty try, had to dig deep and scored further tries through Cammy Hutchison, Pierre Schoeman, and a second from Bennett.
The Parma side’s other scores came from Junior Laloifi, Simone Gesi and David Sisi.
Blair insisted Edinburgh would be much better against Ulster next Saturday.
“We will be right up for the game next week and we will put a performance in,” said the coach.