Braithwaite, who signed for Barca from Leganes on Thursday for €18 million, was introduced for the final 20 minutes and made an immediate impact with a cross for Messi's fourth goal of the game.
The Danish international was also involved in his new club's fifth; Braithwaite's shot was saved by goalkeeper Marko Dmitrovic but Arthur Melo netted the rebound.
"Messi congratulated me [for the pass]," Braithwaite told reporters when asked what the Argentine said during their embrace. "He's a great guy. He tried to make me feel comfortable and looked to find me with his passes.
"I won't wash my kit after the hug he gave me [laughs]. Honestly, I am really happy to have been able to set him up for a goal."
The former Middlesbrough forward, who will only be allowed to play in league matches, was given a warm welcome by home supporters when he replaced Antoine Griezmann.
"The ovation was really special," Braithwaite said. "It's something that I saw on television when I was little, how they treat the new players [here], and suddenly to be here and experience that from the fans is incredible. I am really proud.
"I feel really good here, surrounded by fantastic players. I tried to run and move, which is what I have to do, and my teammates found me. It's been amazing. I am really happy."
Barcelona coach Quique Setien was impressed with what he saw from his new signing and tipped him to make a contribution in the final phase of the campaign.
"Braithwaite did more than I expected, much more," Setien said in a news conference. "Honestly, I think we have made an extraordinary signing. He's going to adapt perfectly to what we need.
"We've seen he has good movement this afternoon, but beyond the goals he was involved in, he's good in space and he understands the game so well. He is an extraordinary player and he's going to help us a lot."
While Braithwaite impressed on his debut, Messi once again stole the show. The Barca captain had gone four league games without scoring, but his four-goal haul against Eibar took his season tally to 23 in all competitions.
With this win, Barca went them two points clear of Madrid at the top of La Liga, although Zinedine Zidane's side can reclaim the lead when they meet Levante later on Saturday.
Setien said the win was "just what was needed" ahead of a testing week, in which his side visits Napoli for the first leg of their Champions League last-16 tie on Tuesday, then face Real Madrid at the Santiago Bernabeu five days later.