The Warriors ran out comfortable 36-17 winners, securing a bonus point before half-time, but gave away two late tries.
It is the third successive match in which Glasgow have slackened after building a commanding lead.
Gregor told BBC Scotland: "We're disappointed by our second-half performance - we need to set ourselves higher standards.
"There were a lot of changes, there were players who hadn't played for weeks and months and a lot of the were coming onto the field at the same time.
"Cardiff played really well, had most of the ball in second half, but we did make it quite easy for them.
"Overall, we're delighted to get the bonus point. I thought Cardiff in the first 20 minutes made it very hard for us, they defended really well, and it took us a while to break them down.
"We had to be very accurate and when we did that it led to tries. We just have to keep on defending and keep on playing.
"We got the bonus point and we made changes (at half-time), so I take that as my responsibility to make sure the team's better than they showed in the second half."
Peter Horne scored three tries in less than 15 first-half minutes, to back up fine displays against Leinster and Zebre.
"He has been excellent for us over the last three years," said Gregor. "He was very, very good for us during the Six Nations, bounced back from being left out of the last two games really well.
"He was one of our best players over in Leinster, and tonight he was great. He's really stepped up as a goal-kicker the last few games, so he's added another string to his bow. He's played 10 for us and Scotland but also showed his worth as a 12."
Scotland head coach Vern Cotter was among the Scotstoun crowd and will have been impressed with Peter's performance before a tweaked hamstring curtailed his involvement.
"It was great, we started to play some good rugby, get ball into our dangerous runners' hands and I managed to sneak on the end of them," said Peter.
"All I can do is keep my head down, work as hard as I can and improve every week. The ethos we have at Glasgow is really helping me and I'm just enjoying my rugby. You've got to earn your jersey back and that's exactly what I'm trying to do."