The County first allowed the Welshmen to increase their overnight 325 for seven to 407 all out, with skipper Chris Cook making 138, before slumping to 76 for five in reply.
Rossington and Saif Zaib launched something of a recovery as the pair put on 126 for the sixth wicket, making 76 and 41 respectively.
But they too then fell in quick succession, and it needed an unbroken stand of 38 between Wayne Parnell and Gareth Berg to see Northants to 251 for seven at the close, still 156 in arrears.
It was the third Championship innings in a row that Northants have suffered an early order collapse, having been 87 for eight and 96 for six in the two innings of the defeat at Lancashire, and Rossington knows it was not good enough again.
“I think there is still a bit of work to do to reach parity," said the skipper.
"It was a fairly disappointing day but if we have a good first session tomorrow, we can get rolling again.
“We found a few interesting ways to get out. It wasn’t the ideal start, we wanted to get in and bat big but that didn’t happen.
“We tried to rebuild and take control, which I think we did a good job of that but another bang-bang in the evening left us a bit behind again."
Rossington was impressed with how Zaib handled himself in a pressure situation, and he said: "It was a tough situation for Zaib to bat but he stood up and did what we asked of him.
“We wanted to bat as long as we could get a foothold in the game and bat as big as we could.
"We had a good partnership going and it would have been good if we could have seen it through to the end of play.”
Glamorgan skipper Cooke said: "I am quite pleased with where we are in the game.
“There is a small boundary on one side and a rapid boundary so it can be hard to stem the flow sometimes with the sweepers.
"It can feel flat but you can go bang-bang pretty quickly as you saw there.
“Rossington and Zaib’s partnerships was a little frustrating but we knew how the pitch would go, we knew there would be partnerships. We tried to keep it tight and took a couple of quick wickets which changes things.”