George Pisi’s late score and Stephen Myler’s successful conversion saw Baxter’s men fall to an agonising 17-16 defeat.
It was hard to take for the Chiefs, who had led 16-10 after Dave Lewis grabbed their only try in the 53rd minute of the match.
Tries from James Wilson and Samu Manoa gave Saints a 10-9 half-time lead, but it looked as though they would suffer a defeat before Pisi popped up to seal the win.
And Baxter said: “Fair play to them, they came down with their internationals out of the side and some injury issues as well and they’ve still forced the win in the end.
“Great credit to them, we just lacked a bit of being direct and clear in what we wanted to do.
“I thought we were a bit passive.
“You look at the scores we conceded in the first half. We had a very strong wind behind us and we gave away silly penalties.
“When we scored we messed up a restart, then we don’t deal with a lose ball, we don’t deal with the breakdown properly. All of a sudden, there’s a try out of nothing.
“It’s frustrating when we look back and 90 per cent of the game is okay. It’s frustrating.
“It’s too many of those odd things that are hurting us when we need to be really well focused and decisive about what we do.
“I’ve said to the guys that we’ve let one slip through our fingers.”