Germany internationals Toni Kroos and Thomas Mueller did the early damage with goals in the opening 21 minutes for Bayern — a beaten finalist in two of the last three years — before Mario Mandzukic sealed victory by bundling in a third in the 77th at Emirates Stadium.
Former Bayern forward Lukas Podolski had given Arsenal hope by making it 2-1 in the 55th but Bayern was well worth the win, and should have scored more in an embarrassingly one-sided first half where the visitors cut through their defensively inept opponents at will.
On current form, perhaps only Barcelona has the beating of Jupp Heynckes' classy team.
"Let's not hide the truth," Arsenal manager Arsene Wenger said, "it will be very difficult. We will try to make the impossible possible."
The second leg is in Germany on March 13, when Porto will look to defend a first-leg lead after beating Malaga 1-0 in Portugal on Tuesday thanks to Joao Moutinho's 56th-minute goal.
Bayern has reacted superbly to the trauma of losing last season's final at Chelsea on penalties at its own Allianz Arena, surging into a 15-point lead in the Bundesliga, qualifying easily from its Champions League group and looking strong thanks to the off-season additions of Mandzukic and defensive midfielder Javi Martinez.
By recently labeling his team as the best in Bayern's history, Heynckes is making a big call. But the Bavarians showed just why they are receiving rave reviews across the continent by outclassing an Arsenal side in the midst of the most trying season of Wenger's 17-year reign.
Winning the Champions League — surely beyond the English club after losing so convincingly here — is Arsenal's last chance of ending a trophy drought that is set to stretch to eight seasons.
"We are currently in astonishing form," Heynckes said. "Our quality is we can punish even the smallest mistakes of our opponents."
Defensively, Arsenal was a mess, with centre backs Per Mertesacker and Laurent Koscielny looking like accidents waiting to happen. The first goal sprung from their lax inter-passing in front of their own penalty box, with Bayern collecting a rushed, errant clearance from Koscielny before Mueller crossed for Kroos to crash home a volley that skidded off the turf and into the net from 18 yards (meters).
Mueller then benefited from more hesitant defending for the second, the lanky forward on hand to poke the ball high into an unguarded net after goalkeeper Wojciech Szczesny parried out a header from an unmarked Daniel Van Buyten.
"We didn't start great and that's been a thing for us this season," Arsenal midfielder Jack Wilshere said. "Maybe it's a bit of nerves or anxiety because we show what we can do when we play."
Bayern sat deeper thereafter, playing within itself in the second half only to break out with a clinical counterattack for Mandzukic's goal. Full back Philipp Lahm provided the overlap and squared a cross for the Croatia striker to score with a fortunate finish that looped up off his heel.
The scruffy strike was hardly in keeping with the visitors' impressive overall display.
"You have to give credit for their class," Wenger said. "In patches in the first half, there was a difference in quality."
With Kroos, Martinez and Bastian Schweinsteiger dominating midfield with their hulking presence and Arsenal's passing awry all over the pitch, the four-time European champions were in cruise control with barely a quarter of the match gone.
Arsenal's players were losing their heads — Bacary Sagna and Mikel Arteta picked up petulant bookings — and they were jeered off at halftime as Bayern supporters belted out another rendition of "Football's Coming Home" in English. Wenger, under serious pressure for perhaps the first time in his tenure, had the look of a beaten man.
Things looked up at the start of the second half, though, and Podolski capped a period of gentle Arsenal pressure by heading into an empty net after Wilshere's inswinging corner evaded a raft of players in the six-yard box — including flapping Bayern goalkeeper Manuel Neuer.
Wilshere was finally making his presence felt and Bayern's early spark faded as Arsenal monopolized possession. Olivier Giroud, on as a 71st-minute substitute, shot straight at Neuer in Arsenal's other good chance.
However, Bayern had the last laugh when Mandzukic finished a sweeping move by bundling in Lahm's low cross at the far post, the ball looping up off the Croatian's heel and beyond Szczesny.
"There's always a chance — we take it to their place, and we need a few goals," Wilshere said. "We have the players to do it, we just have to start better."
Schweinsteiger will miss the second leg after picking up a second-half booking.