Henrikh Mkhitaryan gave the visitors an away goal with the opener in Russia, but a poor playing surface affected the quality of the match and Rostov levelled things up on the night through Aleksandr Bukharov shortly after half time.
Lyon were handed a tricky tie against Serie A high-flyers Roma, but they edged an entertaining and end-to-end tie 4-2 to give themselves the advantage heading into next week's second leg.
Roma's two away goals could prove to be crucial in the reverse at the Stadio Olimpico, but Lyon will travel to the Italian capital with a lead to defend courtesy of goals from Corentin Tolisso, Nabil Fekir and Alexandre Lacazette in the second half.
The most incident-packed tie of the night came in the all-Belgian affair between Gent and Genk as Tottenham Hotspur's conquerors suffered a miserable evening against their domestic rivals.
Just two points separate the teams in the Belgian top flight, but there was a much greater difference on the night as Genk scored five away goals to take complete control of the tie.
Anderlecht are well placed to join their Belgian compatriots in the quarter-finals having beaten APOEL 1-0 in Cyprus, but they were the only team to win on foreign soil this evening.
An all-German clash between Schalke 04 and Borussia Monchengladbach ended all square after Guido Burgstaller had cancelled out Jonas Hoffman's goal, while Olympiacos, who sacked manager Paulo Bento earlier this week, and Besiktas played out the same 1-1 scoreline to give the Turkish outfit a slender advantage.
Europa League last-16 first leg results: APOEL 0-1 Anderlecht, Copenhagen 2-1 Ajax, Rostov 1-1 Manchester United, Celta Vigo 2-1 Krasnodar, Gent 2-5 Genk, Lyon 4-2 Roma, Olympiacos 1-1 Besiktas, Schalke 1-1 Borussia Monchengladbach