Zambrano gave up eight runs in his return from the disabled list Saturday, and Chicago lost 13-3 to the Florida Marlins.
"I was anxious to see that he got through healthy," Quade said. "Our trainers said there were no problems. He's fine."
Zambrano (6-5) made his last appearance for the Cubs on June 30 against the Giants before a sore back forced him onto the DL.
"His velocity was as good as I've seen early on, maybe too good," Quade said. "He was throwing hard (but) probably didn't have the life and location that he's had here when he was on a nice run.
The Cubs right-hander, meanwhile, gave a succinct assessment of his 4 2-3 innings outing.
"I felt good, my fastball was good," he said. "My arm, body (were) good. Intelligence? Not good. Bad pitches to (Mike) Stanton twice."
Stanton, the Marlins' right fielder, hit two home runs, while starting pitcher Javier Vazquez struck out 10.
Stanton had three hits and drove in four runs. He hit a leadoff homer in the fourth inning and added a three-run shot, his 20th, in the fifth that finished Zambrano.
Zambrano (6-5) gave up seven hits and four walks.
Vazquez (6-8) went seven innings. He allowed only one hit through 5 2-3 innings before Aramis Ramirez hit a three-run homer later in the sixth, his 16th.
Now 1-2 against the Cubs this season, Vazquez gave up four hits and walked none.
"Vazquez certainly did a great job, he pitched into the seventh inning with a one-hitter," said Marlins interim manager Jack McKeon. "He did his job, kept us in the ballgame.
"Our offense came alive for a change. We took some bases on balls today, that was key before Stanton hit the home run. Offense played well, Stanton had a great day, Hanley played great."
Stanton's homer in the fifth gave Florida an 8-0 lead. Morrison homered in the sixth off reliever Ramon Ortiz.
Zambrano had been on the 15-day DL since July 1 with lower back stiffness after pitching one inning the previous day in a start against the Giants. He made a rehab start at Class-A Peoria on July 8.
NOTES: With Zambrano's activation, the Cubs optioned reliever Chris Carpenter to Iowa. "He did a great job here, nothing to disappoint us, but given the makeup of the bullpen he needs to go where he can pitch," Quade said. Carpenter had no decisions or saves and a 2.79 ERA in 10 relief appearances ... Marlins SS Emilio Bonifacio's eighth-inning infield single extended his hitting streak to 15 games. ... A day after reaching 1,000 career hits, Hanley Ramirez got a start on reaching his next milestone with his first-inning double to right. Jeff Conine is second in club history with 1,005 hits ... The four-game series concludes at 1:20 p.m. Sunday with Cubs righty Randy Wells (1-3, 6.80) making his 10th start of the season and first against Florida. The Marlins send right-handed Chris Volstad (5-8, 5.40 ERA) who is 0-1 against Chicago after a 5-1 loss on May 19 at Wrigley Field.