The Light Blues’ 0-0 draw at Alashkert for a 1-0 aggregate win sees them into tomorrow’s draw as McAllister’s men put a difficult week behind them to get the job done.
He told RangersTV: "You have got to get in your minds the players are feeling it as well - there are a lot of their mates, teammates and members of the staff not been able to travel, and these issues affect you.
"But we didn't use it as an excuse. I thought we looked together - we might not have played well, but you couldn't point the finger at anyone who wasn't trying - we gave it everything and we came here to keep a clean sheet and win the game.
"We are absolutely delighted that we are in the draw for the group stages. I think the fact that we have now done it four years on the bounce is a fantastic achievement for the players and it is wonderful for the club.
"We are going to be in the draw with some prestigious names - I will hold my hands up, it wasn't pretty tonight, but we have had a difficult two or three days.
"I want to stress, but it was incredibly hot - at the beginning of the game we are talking 33C, and I think it cooled down to 29C.
"We have played better, but we did make the best chances, we hit the woodwork a couple of times and Alfredo [Morelos] has had a couple of good chances where you expect him to score.
"We came here to get a result, and we go back with the result."
Robby McCrorie had a solid competitive debut in goal for Gers, and McAllister continued: "We have loads of belief in Robby and he didn't let us down. I thought he dealt with the ball well and the one awkward shot that came in low and bounced in front of him, I thought he handled that well.
"So it is a very good debut. Nathan [Patterson] too, who hasn't featured a lot, it was another 90 minutes under his belt, so they would be the two big positives."
The Light Blues’ recovery begins now ahead of Sunday’s Old Firm match at Ibrox, and McAllister added on that: "That is one of the reasons why we are spending another night here so we start the recovery now, we can guarantee players will get a good sleep rather than going straight on a plane and we can control their food intake and then get up in the morning and not be in a rush to get back.
"But, we are well aware, when we do get back, we have got to refocus and come together again. We will be bolstered by two or three players when we get back and we look forward to the weekend."
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