Angelino headed Leipzig in front on the stroke of half-time to score his second goal in two games, before Yussuf Poulsen doubled the lead with a sumptuous volley on 66 minutes.
"These are the games you need to win if you want to land in the top four," Leipzig coach Julian Nagelsmann said.
Elsewhere, Lucas Alario's first half header sealed a 1-0 win for Bayer Leverkusen and extended second-from-bottom Mainz's miserable run to four defeats in four games.
Promoted side Stuttgart continued its good start to the season with a 2-0 win at Hertha Berlin, while Freiburg and Werder Bremen shared the points in an eventful 1-1 draw in the south-west.
Marc Oliver Kempf headed in a Daniel Didavi free kick to give Stuttgart an early lead, and Gonzalo Castro fired in the second with a fizzing long-range effort in the second half.
Freiburg took the lead early on through Philipp Lienhart and had a second goal ruled out for offside before Bremen's Niclas Fuellkrug levelled the scores from the penalty spot.
Borussia Moenchengladbach and Wolfsburg shared the points in their late clash after a 1-1 draw.