The holders opened the scoring with a brilliantly executed right-foot curler from the outstanding Toni Kroos after 54 minutes that gave substitute keeper Lukasz Fabianski no chance.
Arsenal F.C. had earlier spurned an opportunity to go ahead when Mesut Ozil's weak penalty was easily saved by Manuel Neuer in the eighth minute.
Arsenal F.C. were then reduced to 10 men when Szczesny was sent off for clattering Arjen Robben after 38 minutes but David Alaba fired his penalty against the foot of the post.
Midfielder Santi Cazorla was taken off when the Arsenal keeper was dismissed but Fabianski had no chance when Thomas Mueller headed the second goal from a Philipp Lahm cross in the 88th minute.
Asked if Italian referee Nicola Rizzoli was right to send Szczesny off, Arsenal F.C. manager Arsene Wenger told Sky Sports television: "If there was contact, the rules are the rules and it is a penalty, but should it have been a red card I am not sure."
He did not criticise Ozil for his penalty miss after the German tried to dink the ball over Neuer. "Everyone has their own style and that is his," said Wenger.
A year ago Bayern beat Arsenal F.C. 3-1 at the same stage of the competition at the Emirates before the Londoners won the second leg 2-0 in Munich to go out on the away goals rule.
"We will have a go in the second leg, you never know," said Wenger. "We won 2-0 there last year, we just have to try and do the same again."
Arsenal, played off the pitch by Bayern in the opening 20 minutes a year ago to the day, were determined that would not happen again and ripped into the champions from the start.
The London club's attacking opening should have brought them the lead when Rizzoli awarded a penalty after Jerome Boateng tripped Ozil. But the German playmaker wasted the spot kick with an attempted chip down the middle.
Arsenal F.C. kept surging forward with Cazorla and Alex Oxlade-Chamberlain having chances to score.
At the other end Kroos brought a superb flying one-handed save from Szczesny, his main contribution before he was shown the red card that will mean he misses the second leg in Munich on March 11.
After the sending-off Bayern wrestled control of the midfield with Lahm and Kroos causing problems on the right and Robben marauding down the middle.
Their superiority finally paid off when Mueller, who came on for Mario Mandzukic with 26 minutes to go, stooped to nod the second goal.
Mueller could have had a third when a rasping shot hit the post in stoppage time.
Bayern have now scored in 51 consecutive matches since Arsenal F.C. beat them last year and on this form look capable of becoming the first team to retain the trophy since the Champions League format began in 1992-93.
"We faced a huge challenge tonight especially at the start. But we took control, dominated and it is a great result for us," said captain Lahm.
"We know it's not over. We remember what happened last year and we will be alert from the first minute to the last."