Desmond's first homer of the season was good for three runs and Griffin outpitched 2015 American League Cy Young winner Dallas Keuchel as the Rangers beat the Houston Astros 7-4 on Thursday night to complete a three-game sweep.
The Rangers have won four straight and have swept three straight home series from the Astros for 10 total games.
Texas leads AL Central at 10-6 following last season's long climb out of a 12-18 hole in early May to win the division title. Rangers manager Jeff Banister downplayed the significance of sweeping last year's wild-card team that finished only two games behind them.
"A sweep of anybody is, obviously, anytime during the year is big," Banister said. "It doesn't matter the name on the front of the jersey."
Beltre had three hits, including his third homer of the season in the seventh inning.
Griffin was another late addition to Texas' roster. He didn't pitch in the majors during 2014-15 following Tommy John surgery and signed a minor league deal with the Rangers in January. He said it was "awesome" to finally feel healthy again.
"I felt like me again," he said. "I felt like I was just able to spin the ball the way I wanted, able to hit my spots better."
Keuchel (2-2) gave up season highs of six runs and 13 hits in six innings. Keuchel didn't allow more than 11 hits in any game last season. The only time that he gave up more than six runs during 2015 was nine at Globe Life Park last September.
Desmond also doubled to go 4 for 11 in the series after previously going 5 for 46, raising his batting average from .109 to .158. A shortstop throughout his previous seven seasons in the majors, he was signed as a free agent on Feb. 29 and moved to left field.
"April's never been my month," said Desmond, who averaged 22 home runs the last four seasons for Washington. "I'm believing in my swing. That's the stuff that I'm looking for. I'm not necessarily worried about hits."
Colby Rasmus homered twice for Houston, giving him five for the season and his ninth career multi-homer game. He drove in three runs and has reached safely in 15 of 16 games.
"There is a sense that they somewhat have our number," Rasmus said. "We have to continue to try to hold our heads up."
Shawn Tolleson earned his sixth save in seven chances.
ANDRUS EXTENDS STREAK
Texas' Elvis Andrus extended his hitting streak to eight games with a third-inning RBI triple to left field that Rasmus misjudged on a head-first dive toward the infield. He's hitting .429 during that span.
Astros: RHP Lance McCullers (shoulder) is scheduled to throw off a mound on Friday.
Rangers: RHP Keone Kela is scheduled for surgery on his pitching elbow that will likely sideline him for three months. To replace Kela on the roster, Texas recalled RHP Phil Klein from Triple-A Round Rock. ... OF Lewis Brinson at Double-A Frisco (shoulder) will miss at least a week.
RHP Collin McHugh (1-2), who has allowed nine earned runs and 21 hits in 12 2/3 innings, will face the Red Sox in Houston. The Rangers will open a three-game road series against the White Sox with LHP Martin Perez (0-1), who has quality starts in each of his three appearances
Padres host the Dodgers to begin 3-game series
Los Angeles Dodgers (106-47, first in the NL West) vs. San Diego Padres (85-68, second in the NL West)
Red Sox look to break 6-game slide, take on the Orioles
Baltimore Orioles (80-73, fourth in the AL East) vs. Boston Red Sox (72-81, fifth in the AL East)
Rangers look to stop slide in game against the Mariners
Texas Rangers (65-87, fourth in the AL West) vs. Seattle Mariners (83-69, second in the AL West)
Phillies look to break road skid, face the Cubs
Philadelphia Phillies (83-69, third in the NL East) vs. Chicago Cubs (67-86, third in the NL Central)
Astros host the Diamondbacks to begin 2-game series
Arizona Diamondbacks (71-83, fourth in the NL West) vs. Houston Astros (101-53, first in the AL West)
Giants take on the Rockies after Davis' 4-hit game
Colorado Rockies (65-88, fifth in the NL West) vs. San Francisco Giants (75-78, third in the NL West)