Postpublished at 22:15
Thanks for joining us on a night that won't go down in Southampton folklore but will be a good memory for Jay Rodriguez.
More live football text tomorrow on the BBC Sport website when Birmingham host Derby. Join us then.
Rodriguez's first goal since March 2014
Crisp low strike from ex-Saint Sparv
Winner joins Spurs & Liverpool in group stages
Second leg is next Thursday
Emlyn Begley
Thanks for joining us on a night that won't go down in Southampton folklore but will be a good memory for Jay Rodriguez.
More live football text tomorrow on the BBC Sport website when Birmingham host Derby. Join us then.
A couple of other results to tell you about. Athletic Bilbao, who beat Barcelona 5-1 on aggregate in the Spanish Super Cup, lost 3-2 in Slovakia tonight to Zilina.
Former Manchester City youngster Robert Mak scored a hat-trick as PAOK Salonika hammered Brondby 5-0.
Southampton boss Ronald Koeman on BT Sports: "We played too slowly in the first half. We played more aggressively and direct in the second half and that was the difference.
"Sometimes you have to play the long ball and play for the second ball. It’s not a good result but we had five or six very good chances.
"In my opinion Steven Caulker’s was a legal and normal goal. It’s never whistled in the Premier League.
"We’re not really the favourites any more as they have a good result. We will fight for a win. Even a 2-2 draw would be good.
"We have to learn - they are a difficult side. We have to look to ourselves and learn from tonight and do things a better way next week."
Southampton defender Steven Caulker had a headed goal disallowed for a foul on a defender.
He told BT Sport: "I think the goal should stand. We were a lot better in the second half. We played a lot more direct and caused them a lot of problems, I’d back us to go through. We’ve got a great team. A 1-0 or 2-0 victory would be nice."
Southampton striker Jay Rodriguez scored his first goal in over a year following injury tonight.
He told BT Sport: "We’re disappointed not to win. We had our chances to win the game.
"I was happy to make it 1-1. It was lovely, I’ve been waiting for it. I want to score as many goals as possible so it’s good to get the first one. With our squad and our team I’m sure we’ll get the win in Denmark."
Fenerbahce beat Atromitos Athens 1-0 with former Manchester United striker Robin van Persie coming off the bench to score in the 89th minute.
That's his first goal for the Istanbul side.
Southampton had many more shots, efforts on target and possession. But there's only one stat that matters. Fouls.
No, of course it's goals. And they got one each.
Not a great result for the Saints, who will now need to score in their second leg in Denmark next Thursday.
Just as I have my full-time entry ready to publish, Southampton come so close to a winner - Maya Yoshida's header is tipped by keeper Johan Dahlin onto the post.
Seconds to go. Doesn't look like either side is going to score a second.
Shane Long is able to limp off with the help of the medical staff so hopefully his injury isn't too bad...
And he's back on now. Crisis over.
Southampton striker Shane Long goes down with what looks like a knee injury, he was going down when Tim Sparv seemed to catch him.
It's not looking good for the Republic of Ireland striker, especially with Euro 2016 qualifiers coming up.
Corner to Midtjylland as Jose Fonte clears Paul Onuachu's cross. But it's headed clear.
Plenty of time for a Saints winner. I'm not sure I remember a huge delay. Seems a lot of stoppage time.
If my very quick internet research is correct Midtjylland sub Paul Onuachu is 6ft 7ins.
Midtjylland make their final change as Paul Onuachu replaces Morten Rasmussen, who actually has Duncan as his name on his back (see 86 mins).
Midtjylland defender Andre Romer is in no rush to take his throw-in and he's booked. Saints boss Ronald Koeman looks irritated.
Midtjylland have a chance to win it as the ball bounces in the box, the Saints defenders have no idea where it is and Rilwan Hassan tries to control it. The ball breaks on the edge of the box to Morten Rasmussen but his shot is deflected.
Rasmussen is known as Duncan apparently because he looks like ex-Everton and Rangers striker Ferguson.
Southampton bring off England Under-21 midfielder James Ward-Prowse, who has been very good today. His set-pieces have caused lots of problems for the visitors. On in his place is Spanish summer signing striker Juanmi.
The Saints are going for this now.
James Ward-Prowse almost gives them that goal with a curling free-kick from 25 yards which goes just wide. And that's his last action...