Dan Ndoye put the hosts in front on 35 minutes minutes after Taulant Xhaka was sent-off for two bookable offences.
Adam Szalai added a second three minutes into the second half but the Crues produced a disciplined performance against their much-fancied opponents to keep the tie alive ahead of next Thursday night's return leg at Seaview.
"Overall I guess we're slightly disappointed we've lost 2-0," Baxter told the club's Facebook page.
"The two goals were probably avoidable in many ways.
"We haven't cleared the first ball from the edge of the box, which we'd been doing for 30 minutes.
"Then straight after half-time a ball comes into the box and we don't defend the near post well enough.
"The boy has flicked it through his legs, which is lucky from his point of view.
"So they are two goals which I'm disappointed in conceding.
"We'll take nothing away from this team though because they've given us a chasing around this football pitch tonight and we have stayed with them every inch of the way.
"We created a couple of brilliant chances of our own in the last ten minutes.
"Johnny McMurray gets a header away, which by his own admission he thought he was going to score, so that's a highlight for us.
"I thought the team overall in their discipline, their shape, their effort and the hunger and pride they played with in front of an unbelievable noise at this magnificent stadium and were excellent.
"We've come here and we haven't embarrassed ourselves in any shape or form, in fact we've done our club proud and our families proud.
"I'm delighted for everyone connected to the football club because after conceding a second goal so early we could have folded.
"Our boys were out on their feet in 30 degree heat here, but they stuck to the task.
"2-0 could have turned to three, four or five, but they worked tirelessly to a man and I'm so pleased and proud of them.
"We'll go back to Seaview, we've nothing to lose and we'll give them a Seaview roar, we'll put it up to them and see what they've got."