The 2-0 Europa League brought an end to AC Milan's 13-match unbeaten run while Dortmund slumped to a 2-1 defeat to Salzburg.
Under-fire Arsenal manager Arsene Wenger was relieved after his side's impressive 2-0 Europa League last 16, first-leg victory at AC Milan on Thursday, following a "nightmare" week.
Henrikh Mkhitaryan drove Arsenal into a 15th-minute lead with his first goal for the club, and a shellshocked Milan fell further behind before half-time as Aaron Ramsey waltzed through to score.
Wenger desperately needed a positive result after coming under pressure for a run of four straight defeats, and he got just that against a Milan side who saw a 13-match unbeaten run ended.
"It is an important win of course after we had a nightmare week," the 68-year-old Frenchman told BT Sport.
"I am happy with the spirit and response we gave. When you are knocked down like a boxer you don't have time to react and you are only halfway up, but you have to respond.
"Your pride and desire to show quality has to come through in the game."
Gennaro Gattuso's seven-time European champions now have it all to do ahead of the second leg at the Emirates in seven days' time.
"Arsenal are in a very difficult period, but are a great team, but we made too many mistakes," said Gattuso.
But Arsenal, 13 points adrift of the top four in the Premier League, will be confident of continuing their bid to return to the Champions League by winning Europe's second-tier competition for the first time.
"Overall it is important to win, but it is not qualification. We have to finish the job at home," Wenger added.
Arsenal started brightly at the San Siro but were lucky not to concede a penalty when Hakan Calhanoglou stayed on his feet despite appearing to have been fouled by onrushing Arsenal goalkeeper David Ospina.
Just moments later Arsenal broke the deadlock, as Mesut Ozil fed Armenian Mkhitaryan, a January signing from Manchester United, to cut inside and fire in a shot which deflected past home keeper Gianluigi Donnarumma off Leonardo Bonucci.
The Gunners grew in confidence after that strike, with Donnarumma making two good saves late in the opening period to deny Calum Chambers and Danny Welbeck, before Mkhitaryan hit the crossbar.
But the visitors did double their advantage before the interval, as Ramsey ran onto Ozil's perfectly-weighted through ball and rounded Donnarumma to tap into an empty net.
Milan improved in the second half, but wasted their best chances, failing to truly test Ospina.
Giacomo Bonaventura scooped a shot over when he should have done better, while Suso drilled a left-footed strike out for a throw-in.
Colombian keeper Ospina did well to beat Nikola Kalinic to a through ball when the substitute looked to be in on goal, as Arsenal closed out a much-needed win.
Salzburg shock Dortmund
Borussia Dortmund slumped to a shock first-leg 2-1 loss at home to Salzburg, although Andre Schurrle's second-half strike did cut the deficit ahead of next week's return game in Austria.
Kosovo international Valon Berisha scored twice shortly after half-time to put Salzburg in control, the first from the penalty spot, and leave German giants Dortmund facing a surprise exit.
La Liga title challengers Atletico Madrid eased to a 3-0 victory over Lokomotiv Moscow at the Wanda Metropolitano.
Diego Simeone's side, who were dumped out of the Champions League in the group stage after finishing behind Roma and Chelsea, put themselves in a dominant position thanks to goals from Saul Niguez, Diego Costa and Koke.
Spanish international Costa has now scored six goals for Atletico since returning to the club from Chelsea in January.
Brazilian defender Marcelo scored the only goal as last season's semi-finalists Lyon grabbed a 1-0 win at CSKA Moscow, as they look to go one better this term with the final at their Groupama Stadium.