Bianchi went 2 for 4, entering the game as a pinch-hitter in the first for Jean Segura, who was hit in the head in the dugout by a bat.
A TV replay showed the All-Star getting hit on the right side of the head during a warmup swing by teammate Ryan Braun, who was facing the field on the top steps of the dugout while Segura was coming up behind him.
Manager Ron Roenicke said that Segura didn't have a concussion or fractures, but needed stitches to close a gash on his face. It was unclear how long he might be out.
Bianchi drove in two runs with a bases-loaded single in the second off Cubs starter Travis Wood (1-3).
Estrada (2-1) had nine strikeouts.
Francisco Rodriguez tossed a 1-2-3 ninth for his 11th save.
Estrada retired the first 10 hitters he faced - a few on fly balls deep into the outfield - before Luis Valbuena turned on a 2-0 fastball for a solo homer to right.
The righty followed with another stretch of 10 straight retired, effectively mixing a changeup and curve with fastball in the high 80s.
He maintained control until giving up a two-run shot to Welington Castillo in the eighth to get the Cubs within two. Estrada left after 7 2/3 innings, allowing four hits and no walks.
Chicago, which lost its fourth straight, is already 10 1/2 games back in the NL Central behind division-leading Milwaukee, which has the majors' best record at 18-6.
The Brewers have won seven of eight, taking games of late with a formula of pouncing on opposing starting pitchers early to make life easier for their pitchers. Estrada and his fellow starters have held up their end with a sub-2.00 ERA in their past six starts.
But Estrada's outing was overshadowed by the injury to Segura, who had to be helped to the clubhouse after accidently getting hit while Braun was wheeling the bat behind his back to his left. The injury added to what's already been a tough start of the season for Segura, who was hitting .239.
Manager Ron Roenicke had dropped him to eighth in the order for a few games to try to get Segura going. Segura was back in the second spot in the lineup against the left-handed Wood before departing without getting an at-bat.
Segura also missed the last two weeks of spring training with sore right shoulder.
Braun finished 1 for 4 with a bunt single in the seventh before being lifted in the top of the ninth defensively for Elian Herrera. Braun, has been dealing with a lingering right hand injury, left because off a slight muscle strain in his chest that had flared up on Friday night.
Khris Davis hit a solo homer in the sixth inning for the Brewers.
Notes: Chicago placed struggling reliever Jose Veras on the 15-day disabled list with a strained left oblique before the game. RHP Brian Schlitter was recalled from Triple-A Iowa, and arrived in time for the game. Veras, who had a 15.88 ERA and blew both of his save opportunities. ... The Cubs have lost seven of eight to Milwaukee dating back to last season. ... Rodriguez picked up his 315th career save, three behind Rick Aguilera for 17th place on the all-time list. ... Chicago will try to avoid a sweep by sending Jason Hammel (3-1) to mound Sunday against Wily Peralta (3-0).
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