Visiting head coach Franco Smith had made 10 changes from the side that suffered a heavy Investec Champions Cup defeat by Leinster last time out, but their struggles continued in Parma.
They failed to score in Dublin and were to go another 42 minutes before they managed to get the scoreboard ticking in Italy.
After some untidy play by both sides, fly-half Da Re had given the hosts an eighth-minute lead with a penalty in front of the posts.
Zebre thought they had the opening try when Giovanni Licata went over under the posts, but the replay showed the number eight had lost hold of the ball before touching down.
Moments later, centre Scott Gregory also let the ball slip out his grasp as he crossed the line, but the home side did stretch their lead on the half-hour mark through another Da Re penalty.
Glasgow’s 24-match winning run over Zebre was seriously under threat as the second half began.
However, home lock Matteo Canali was yellow-carded for a head-first challenge on Sebastian Cancelliere and Glasgow took immediate advantage, with full-back McKay finding the corner and Tom Jordan adding the conversion.
McDowell added another try under the posts seconds before Canali’s return, with Jordan kicking the extras to make it 14-6.
Zebre would hold their own again with both sides back to full strength, but the damage had been done.
Glasgow Warriors head coach Franco Smith: “We know how much Zebre have improved.
“They slowed down our breakdowns slow made it difficult for us to create momentum, so all credit to them.”
Zebre: Prisciantelli, Trulla, Paea, Mazza, S Gesi, Da Re, Fusco, Fischetti, Di Bartolomeo, Hasa, Canali, Krumov, Ruggeri, Stavile Bravin, Licata.
Replacements: Bigi, Buonfiglio, Pitinari, Nasove, Ferrari, Garcia, Morisi, Gregory.
Glasgow Warriors: McKay, Cancelliere, Smith, McDowall, Steyn, Hastings, Dobie, Bhatti, Hiddleston, RIchardson, Oguntibeju, Samuel, Ferrie, Vailanu, Mann.
Replacements: Matthews, Schickerling, Talakai, Williamson, Darge, Kennedy, Jordan, Cordero.
Referee: Peter Martin (IRFU).
