Texas manager Ron Washington said he spoke with the team after the game, but didn't divulge his message.
"I thought they needed to hear from the manager," Washington said. "We're not the first team to lose six in a row and we probably won't be the last."
Lind hit a three-run homer, and J.P. Arencibia added a two-run shot for Toronto. The Blue Jays outscored the Rangers 24-4 to sweep a four-game series at Texas for the first time.
"I keep hearing the Rangers are struggling and all that," Gibbons said. "Maybe we're pitching good. Maybe we're playing good. I get tired of hearing that. I tip my hat to my guys."
Derek Holland (5-4) allowed four runs and 10 hits in six innings for the Rangers, who have only scored eight runs in their slide.
Elvis Andrus had an RBI single, and Ian Kinsler walked with the bases loaded to break up the shutout in the ninth.
Texas has lost six in a row at home for the first time since a seven-game slide from July 20-Aug. 3, 2005.
"I didn't sense any frustration," Texas outfielder David Murphy said. "We're past that point. Things didn't happen. With the amount of talent we have here, we're very close to turning this thing around."
Arencibia, who entered the series in a 3-for-41 rut, hit his second homer in three games. He lined an 0-1 pitch just over the 14-foot fence in left, a two-run shot that gave the Blue Jays a 2-0 lead.
It was the first home run Holland has given up since May 19.
Rasmus led off the fourth with a drive off the foul pole in right for his 13th homer.
The Blue Jays scored four times in the seventh to extend their lead to 7-0. Jose Bautista had an RBI single, and Lind capped it with a three-run homer to right off Kyle McClellan.