Well, it's not like a completely new experience. After all, he began this season with eight career saves.
Mostly, though, he has been the guy the Yankees counted on to get them through the eighth inning and hand the ball to Mariano Rivera.
Apparently, old habits die hard.
"In the past whenever I've had to close a game, I forget that I've got to go find the catcher and shake hands," Robertson said. "It's just different."
He laughed about this a few times Saturday afternoon after getting his first Yankee Stadium save as The Guy Who Is Replacing Mariano Rivera.
All he had to do was run through a stretch of the Angels' lineup that has Mike Trout, Albert Pujols and Howie Kendrick.
Trout singled with one out and stole second to put himself into scoring position. But Robertson got Pujols on a fly to left and struck out Kendrick to end a 4-3 victory.
It was Robertson's first save in 20 days, after a stay on the disabled list, and his return helps stabilize a bullpen that has had a stressful week.
Yankee starters didn't make it through the fifth inning in three out of the last four games, including the pine-tar contest in which Michael Pineda was ejected in the second inning.
That's probably what made Saturday afternoon's game so satisfying for the Yankees. Well, worrisome and satisfying.
Yankees starter Vidal Nuno, who is replacing injured Ivan Nova in the rotation, lasted just 4 1/3 innings and allowed three earned runs.
But Yankees manager Joe Girardi mixed and matched four relievers for 4 2/3 scoreless innings, including Robertson's ninth inning.
To Girardi, a win is a win is a win. Style points are irrelevant as the Yankees (14-10) remain in first place in the American League East.
Now about that rotation. Masahiro Tanaka is off to a nice start. CC Sabathia has shown flashes of being plenty good enough. Hiroki Kuroda had his first bad start Friday.
So that leaves Nuno and Pineda to fill in the back of the rotation, and the Yankees have no idea what to expect.
All that's obvious is they need a bit more than the 4 1/3 innings they got from Nuno on Saturday.
"It wouldn't be such a big deal, but we lost Michael so early in that game," Girardi said. "That's the one that we really had to use our bullpen on. We have an off-day Monday and I'll have [Adam] Warren back [Sunday]. We'll see about some of the other guys. We'll be OK."
It's perhaps the greatest tribute to Robertson and the confidence the Yankees have in him they were completely comfortable moving him into the closer's role. In fact, their internal discussions were not about who would replace Rivera, but who would replace Robertson.
Amid the scramble to find innings, 26-year-old right-hander Dellin Betances has emerged as a potentially dominant reliever.
Once upon a time, he was one of the brightest prospects in the Yankees system, but injuries and inconsistency seemed close to derailing his career.
He's a reminder that young players don't come with guarantees or timetables. He has methodically worked his way back into the picture and now seems close to establishing himself as an important bullpen contributor.
"For me, every time I go out there the experience has helped me," Betances said. "A lot of the guys have given me good advice. It's having the confidence and faith. Thank God, everything has gone well."
He's 6-8, 260 pounds, and an imposing sight on the mound. His fastball has averaged 95 mph this season, and his curveball is workable as well.
"I'm happy for the kid," Girardi said. "He's been through a lot. He's had to change roles (moving from the rotation to the bullpen), and he's had to fight to get here. He's thrown the ball extremely well for us. Probably today was the toughest situation we've put him in so far. He responded extremely well."
The Yankees aren't a young team. The first eight guys in the order on Saturday averaged 35 years and 13 years of big league experience.
But at the bottom, 23-year-old catcher John Ryan Murphy hit his first career home run and drove in three runs. And then Betances, another product of the Yankee farm system, got four big outs as Girardi worked to get the ball into Robertson's hands.
All in all, it was a good day at the office.
"We go through stretches like this all the time in baseball," Robertson said. "The next thing you know, you'll get a full nine innings out of a starter, and the game's over in 2 hours, 40 minutes. It happens."
Richard Justice is a columnist for MLB.com. Read his blog, Justice4U.