Drew needs surgery and will undergo tests on Thursday to see what the extent of the injury is.
He was injured trying to score from second base on a double by Chris Young to left field, when Ryan Braun's relay to Casey McGehee beat him on a feet-first slide at the plate, which Jonathan Lucroy was blocking. Drew, who ran through a stop sign from third-base coach Matt Williams, caught his foot on Lucroy as he slid past the catcher.
"Home plate, it's a dangerous place where collisions happen and slides happen and there's a lot of blocking and stuff like that," Gibson said.
Drew couldn't get up and was helped off the field by two trainers.
"He's a shortstop," infielder Ryan Roberts said. "That's the captain of the infield right there. Look around the league and all the big-name players play shortstop, so it's a big loss for the team and we'll find a way to bounce back from it."
Drew was hitting .252 on the season with five home runs and 45 RBIs. He recorded a .980 fielding percentage and made seven errors.
Gibson said he didn't have any answers on filling Drew's spot, but veteran Willie Bloomquist figures to see action at shortstop in the immediate future.
"We're certainly going to miss him," Bloomquist said. "He's a huge part of this team. We're going to have to find a way to pick up the slack, and I'm going to have to figure out a way to elevate and do things a little bit sharper and a little bit better."
In his four full years manning the middle of the D-backs' diamond, Drew has played 150-plus games three times and has twice had to go on the 15-day disabled list -- in 2006 and '09 -- both hamstring injuries.
He has a .269 career average with 70 home runs, 321 RBIs and a .978 fielding percentage.
"We now know what the Giants are feeling," Gibson said, referring to the season-ending injury to San Francisco's star catcher Buster Posey earlier this year. "We're in a similar situation [with] this and we'll have to find a way to get through it. There's nothing we can do about it.
"We wish him well and hope that he has a good recovery and he'll be able to resume his career next year."