The visitors came back strong in the second half, with Cristiano Ronaldo grabbing his first two minutes in the second half.
After a hard loss, Bayern manager Carlo Ancelotti was quoted to be saying by Reuters, "Small details have decided the game".
Bayern should have been 2-0 up at the break, but Arturo Vidal, who headed the hosts into a first-half lead, skied a penalty.
Ronaldo reached 50 goals in April 2013 after 96 matches that was despite only scoring once in his first 32 European appearances for Sporting CP and United.
"We played well and won well, but everything remains open for the second leg. Neuer?" Going a man down didn't make it easier to say the least, but we still did quite a good job.
Diego Costa could be sold
Tianjin, coached by Italy's World Cup-winning former captain Fabio Cannavaro, won promotion to China's top flight in October. It's safe to say that we have the same transfer targets as those illustrious clubs".
Real drew level two minutes after the break when Carvajal fired in a cross from the right which Ronaldo volleyed past Manuel Neuer from close range.
But a touch of Ronaldo genius is always just around the corner to remind us just how great an athlete and footballer he really is.
"We hope he is back with us in a short time". In the end, both teams came away from their Champions League quarterfinal match relatively satisfied and looking ahead to next week's second leg in England.
Yet Real remain at La Liga's summit and with the upper hand in their Champions League quarter-final, proving that even a Ronaldo performing below his best may be enough to win some of the biggest accolades in European football.
However, Bayern chairman Rummenigge is sure their star striker will be ready to play in the return fixture at the Santiago Bernabeu next Tuesday.
French presidential hopefuls wrap up campaigns in tight race
Reports suggest a statement was written to the studio which said the European Union "did great harm" to France and its citizens. Opponents of Le Pen and her anti-immigration National Front party also skirmished with police outside a Paris rally this week.
Ronaldo had scored twice in his previous eight Champions League games and had failed to find the net in any competition for a month but on Wednesday he netted twice either side of halftime to end the German champions' run of 16 straight home wins in the competition.
The Spaniard hit a brace, with one of those coming in injury time to keep Madrid's title dreams very much alive.
A Paulo Dybala double ensured Juventus took a massive step towards the semi-finals after a commanding win over Barcelona in Turin.
It had been a frustrating first half for the Portugal forward and his team-mates, Bayern controlling possession and ceding few opportunities for Madrid to attack.
2 officers killed in Paris attack; shooter 'neutralized'
After the loud bangs, she says: 'A sudden noise where we were was mistaken for gunfire and had police shouting us off the street. The attacker is believed to have deliberately targeted the officers, according to reports, and was known to security services.