Ibanez hit a leadoff homer in the second and Chris Iannetta had a long double later in the inning.
Darvish gave up two runs on three hits and a walk in three innings.
Darvish didn't strike out anyone. He led the AL in strikeouts last season.
''He settled down and made his pitches,'' Texas manager Ron Washington said. ''We got three innings out of him.''
Angels starter Tyler Skaggs, a solid candidate for the back end of the Angels' starting rotation, allowed three hits and two runs with two walks in 2 1-3 innings of his spring debut.
Sometimes, managers try to hold back pitchers in spring training, rather than have them face division rivals.
''At this stage, teams are just getting their work in,'' Angels manager Mike Scioscia said.
Ibanez and Mike Trout each got two hits for the Angels.
The Rangers-Angels game was the only one in spring training Tuesday where Major League Baseball's expanded replay system was being tested.
There was one call that got reviewed, and the umpire's ruling was confirmed. J.B. Shuck of the Angels was safe after stealing second with a headfirst slide in the seventh.
The review took 90 seconds.
''It won't always be that easy,'' Texas manager Ron Washington said.
Rangers: Darvish, 13-9 with a 2.83 ERA with Texas in 2013 was happy with his second start of the Cactus League season.
''I think it was good overall because I was able to try a few things,'' Darvish said through a translator.
Darvish said he and catcher Robinson Chirinos both wanted a fastball on the pitch that Ibanez hit for the home run.
''But I threw a four-seam fastball just to see if I could get a popup. But it was a huge popup,'' Darvish said.
Trout singled and lined out against the Texas ace. Darvish said he threw only fastballs to the Angels star.
''I didn't want to show him anything else. I think the approach toward Trout is going to be a little bit different this year,'' he said.
Angels: Skaggs allowed both his runs in the third after breezing through two innings.
''I was flying open,'' he said. ''I was worried about the runner too much and that caused my fastball to rise up too much,'' he said.
Skaggs began his career in the Angels' organization and went 3-6 with a 5.43 ERA in 13 career games with Arizona over the past two seasons.
After being drafted in the first round by the Angels in 2009, Skaggs was dealt to Arizona a year later in a multiplayer deal that sent Dan Haren to Anaheim. Skaggs returned to the Angels this past offseason in a three-team deal that moved Mark Trumbo to Arizona, and has hopes of claiming one of the two open spots in the Angels' rotation.
''His stuff is real,'' Scioscia said. ''He's ready for the challenge. I know he wants it.''
WAITING ON ALBERT
Angels slugger Albert Pujols was hitless in three at-bats and now is 0 for 8 in the Cactus League season. He bounced out on Darvish's first pitch in the first. He also twice flied out to center field.
Angels star Josh Hamilton said he wouldn't put a timetable on his return to the lineup. He left camp one week ago with a left calf strain and has been hitting and throwing but no running. He's expected to be out at least one more week.