Marcel Sabitzer's quick-fire double in the first half put Erik ten Hag's team in control of the tie, but Sevilla, record six-time Europa League champions, pulled one back in the 84th minute when Jesus Navas' shot pinged off the feet of Tyrell Malacia and keeper David de Gea into the net.
United were reduced to 10 men when defender Lisandro Martinez was carried off injured, and Youssef En-Nesyri's stoppage-time header glanced off Harry Maguire into the goal to give Sevilla a huge lift ahead of next week's second leg.
"I think we had the game in hand," Ten Hag told BT Sport. "We should have scored three or four ... and then some unlucky moments with injuries. And then we conceded two own goals and that's bad luck. We have to deal with it. Of course, we have to learn that we have to kill the game."
Sabitzer, who is on loan from Bayern Munich, netted his first European goal for United in the 14th minute, when he controlled a pinpoint pass from Bruno Fernandes and shot past keeper Bono.
The Austrian doubled his tally seven minutes later, latching on to a fine pass from Anthony Martial and finishing well.
Sabitzer, however, said that it was tough to find words after his team's late meltdown. "A rollercoaster," Sabitzer said. "We had to finish the game off, it's a draw, was not what we wanted, have to go to Seville and get it down there.
"We had the chances, we were (confident) in the game and I think we controlled the game, but unfortunately we gave easy goals away."
Martinez appeared to suffer a serious ankle injury and was carried off the pitch by two Sevilla players who are his Argentina teammates.
United, bidding for their second trophy of the season after winning the League Cup, thought that they had scored seconds after kick-off, but Jadon Sancho's effort was disallowed for offside.
Sevilla, languishing in the bottom half of La Liga, were outplayed in the first half but they nearly got on the scoreboard seconds before the interval when De Gea stopped Tianguy Ninazou's close-range header and Raphael Varane cleared it off the line.
