Tillman, 24, scored a play-off career-high 22 points and grabbed 13 rebounds as Memphis overcame the absence of injured Ja Morant.
The Denver Nuggets beat the Minnesota Timberwolves 122-113 to go 2-0 up.
In an attritional game, the Lakers’ Anthony Davis suffered a cut eyelid while Lebron James was drawn into a verbal back and forth with the Grizzlies’ Dillon Brooks – who was not in the mood to apologise post-game.
“I don’t care, he’s old,” Brooks said of 38-year-old James. “I poke bears.”
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The Lakers won the first game of the best-of-seven series on Monday, while the next game will be played on Sunday in Los Angeles.
The Bucks were missing the injured Giannis Antetokounmpo but had enough to overcome the Heat in their Eastern Conference play-off as seven players scored double figures.
The performance of the night arguably came from the Nuggets’ Jamal Murray, who scored 40 points for the top-ranked side in the West.
The 26-year-old missed the whole of last season with torn knee ligaments.