The patience paid off Sunday.
Fielder and Moreland homered during Texas' four-run fourth inning, and the Rangers beat the Pittsburgh Pirates 6-2 for their eighth straight series victory at home.
"They know that the stroke is still there, they know that the power is there," Banister said. "I believe in the impact that they have on our lineup, and our players in that clubhouse believe in the impact in those players and what they're going to be able to do for us this year."
Fielder ended a career-long homer drought at 34 games with his third of the season leading off the fourth. Moreland, who stopped a 1-for-27 skid with a home run Saturday, hit a three-run shot halfway up the upper deck in right field for his sixth, making it 4-2.
"You can't really focus on the numbers," Moreland said. "That's for everybody else to worry about. You've got to go up there and continue to focus on your work and try not to base it on the results. That's what I've been trying to do."
Texas starter Martin Perez (3-4), with a history of getting rattled in jams, escaped an inning with a runner at third and no outs for the second consecutive start. The left-hander gave up two runs in six innings and finished May with a 2.23 ERA.
Pittsburgh lefty Francisco Liriano (4-4) had yet to give up a homer to a left-handed hitter this season. Two of his four walks came before Moreland's homer, and he has nine walks in his past two starts.
"There's times when he gets in those situations that he seems to overthrow," manager Clint Hurdle said. "Two runaway walks to right-handers and two missed locations to left-handers. They both made us pay."
Texas closer Sam Dyson, pitching in a non-save situation for the second straight day, capped three scoreless innings from the bullpen with two strikeouts in the ninth.
The Rangers (29-21) moved a season-high eight games over .500 and took over first in the AL West, a half-game ahead of Seattle after the Mariners were swept by Minnesota.
Perez twice escaped without allowing a run when the first two hitters reached for the Pirates, who dropped their second interleague series of the season.
Francisco Cervelli's single in the fourth was followed by a double from Sean Rodriguez, but Perez got Matt Joyce and Chris Stewart on groundouts that kept the runners in place before throwing a called third strike to Jordy Mercer.
After Starling Marte's walk and another single from Cervelli to start the sixth, Rodriguez flew out, Joyce struck out and Stewart grounded out.
"Now I'm more comfortable with runners on base and no outs," said Perez, who is 6-1 with a 2.15 ERA in his past 11 home starts. "I just need to throw my pitch where I want and not try to do too much."
Jurickson Profar, who had two singles, has a hit in all nine of his career starts in the leadoff spot for Texas. He is hitting there and playing second during Rougned Odor's seven-game suspension for punching Toronto's Jose Bautista during a May 15 brawl.
Profar also scored from second in the seventh without the ball touching the bat. Reliever Rob Scahill's pitch went to the backstop on Adrian Beltre's swinging third strike, and Stewart's throw to first was wide. Profar didn't stop and slid past Stewart's late tag.
The eight straight home series victories are the most for Texas since taking eight in a row in 2011-12. The Rangers last won nine straight home series in August-September 1993.
Pirates: LHP Jeff Locke (3-3, 5.08 ERA) goes in the opener of a four-game series Monday night at Miami. Locke has won three of last four decisions and is 9-5 with a 3.31 ERA in 25 starts against NL East teams.
Rangers: LHP Derek Holland (3-4, 5.21 ERA) pitches on six days' rest in the opener of a three-game series at Cleveland. Holland is 0-3 in May but has a 2.84 ERA in his past two starts, when the Rangers scored just one run.