A graduate of the La Liga club's famed youth academy, 22-year-old Bartra's chances have been limited since he joined the first-team squad last season but he was deployed alongside Gerard Pique in place of the injured Javier Mascherano for the Group H clash in Glasgow.
Barca spent the close season trying to sign another centre back as a long-term replacement for captain Carles Puyol, who is poised to make his comeback after knee surgery, but Bartra's polished display in Scotland suggested he may be the answer.
He coped efficiently with Celtic's most dangerous threat, combative Greece international Giorgios Samaras, and did not commit a single foul during the match, even striding forward in the second half and unleashing a powerful long-range drive that forced a sharp save from Fraser Forster.
His eye for goal will help his case for a regular place in Martino's side and he scored from Lionel Messi's assist in this month's 4-1 win at home to Real Sociedad in La Liga.
"He played an impeccable match," Martino said at a news conference.
"He was up against their most difficult player (Samaras) and he did a very good job," added the Argentine.
Although Barca were unable to extend their lead against a team who played the final half an hour with 10 men - Celtic captain Scott Brown was dismissed for kicking out at Neymar - there was general satisfaction with their showing.
The Catalan club, bidding for a third European crown in six years after triumphs in 2009 and 2011, top the group on six points from two matches, two ahead of AC Milan after the Italians were held to a 1-1 draw at Ajax Amsterdam.
They have also made a fine start to their La Liga title defence, amassing a maximum 21 points from their opening seven games to top the table along with Atletico Madrid.
A battling Celtic twice threatened the Barca goal after losing their captain when Victor Valdes produced a brilliant reaction save to deny James Forrest and Charlie Mulgrew headed narrowly wide from the resulting corner.
"It was a pretty complete performance and we played with a lot of consistency," said Martino, who is making his debut in Europe's elite club competition this season.
"Apart from the Valdes save we had almost no problems," he added.
Barca's next outing is a La Liga game at home to Real Valladolid on Saturday when they will again be without injured World Player of the Year Messi.
The Argentine, who netted a hat-trick in the 4-0 win at home to Ajax on matchday one, should be back for the home and away games against Milan on Oct. 22 and Nov. 6 respectively.