Ten-man Bremen looked set to claim a goalless draw after a stubborn defensive display, with goalkeeper JirI Pavlenka in fine form, but Niklas Sule's deflected strike sealed the points for Bayern in the 75th minute.
With Borussia Dortmund not in action until Sunday, Kovac's side hold a four-point lead at the top of the Bundesliga, with just four fixtures remaining.
And the former Eintracht Frankfurt coach – whose side scored 27 goals in the six Bundesliga games preceeding Saturday's encounter – was thrilled that his side's patience paid off.
"It was difficult for both teams today, the weather conditions were extreme," Kovac told Bayern's website.
"From freezing, so to speak, to 25 degrees, that's not easy. We knew that Bremen is very difficult to play. So we had to play patiently.
"The red card has certainly made things easier for us in the second half. I would have wished that we make more of the chances that we had.
"But 1-0 is enough, we have three points and are happy that we can celebrate Easter as leaders. The victory is fully deserved"
Serge Gnabry and Robert Lewandowski both drew a string of fine saves out of Pavlenka before Milos Veljkovic received a second booking.
Thomas Muller squandered a golden chance shortly after, but Sule was on hand to save the day, and the defender joked that – after two wayward strikes in the 5-0 victory over Dortmund on April 6, he was due a slice of luck.
"I cried out at the edge of the box so Thiago [Alcantara] saw me," Sule said.
"I already had two shots against Dortmund, but they went into the safety fence at the edge of the field. That's why it was nice that today I could decide the game with the deflected shot."