The result leaves the tie hanging in the balance as the Red Devils head back to Old Trafford next week, with their hopes of salvaging a dismal season still alive but far from certain.
Manchester United, mired in 14th place in the Premier League and out of every other competition, have pinned their faint hopes of Champions League football on this tournament.
They took the lead just before the hour mark through Joshua Zirkzee, who ended a five-game goal drought by bundling in a pinpoint pass from Alejandro Garnacho.
But their advantage lasted just 12 minutes, undone when captain Bruno Fernandes conceded a penalty for handball. Oyarzabal, the man who fired Spain to Euro 2024 glory, stepped up and coolly sent Andre Onana the wrong way in the 70th minute.
"We had control until that penalty, it shifted everything," Manchester United boss Ruben Amorim said post-match. "Now we take it to Old Trafford, and we'll fight to win there." Despite the setback, Amorim insisted his injury-hit squad, down to just 18 fit players, are desperate to turn things around. "They're hungry to score and win games," he added.
Real Sociedad, sitting ninth in La Liga and dreaming of lifting the Europa League trophy at rivals Athletic Bilbao's San Mames, pushed hard for a late winner. Substitute Orri Oskarsson squandered two golden chances, first firing wide and then being denied by a sharp Onana save.
"Those misses hurt, but we're still in this tie, we'll go there with fire," Oyarzabal said, reflecting on a game where his side struggled early but grew into it after his equaliser.
The first half had been a cagey affair, with neither team finding their rhythm. Manchester United edged it, though, as Fernandes and Zirkzee both saw efforts blocked by Real Sociedad's resolute Aritz Elustondo.
After the break, the Red Devils upped the tempo, Garnacho skewed a shot into the side-netting before linking up with Zirkzee for the opener. "We've got to be ruthless with our chances - we had enough today," Zirkzee said after the match. "I'm happy to score, but we needed the win and didn't get it."
Real Sociedad, boosted by the return of Oyarzabal, Takefusa Kubo, and Luka Sucic after missing their 4-0 La Liga loss to Barcelona, found life after the penalty. Brais Mendez tested Onana with a curling strike, but it was Oskarsson's late profligacy that kept the scoreline level. "We didn't suffer too much defensively, but creating chances was tough until we scored," Oyarzabal admitted.
Amorim, who on Wednesday downplayed the "crucial" nature of this competition while hinting it could rescue United's season, now faces a must-win second leg. The victors will meet either Lyon or FCSB in the quarter-finals, but for now, all eyes are on Old Trafford.