Nothing has changed in that regard since he got the 'C' on his sweater.
Tavares put the Islanders in front late in the second period, and New York overcame a quick two-goal spurt by Edmonton's Taylor Hall for a 3-2 victory over the Oilers on Thursday night.
New York (3-2-2) has played six one-goal games this season and this time maintained its slim advantage throughout a scoreless third period against the skidding Oilers (1-6-1).
The 23-year-old Tavares, who became captain last month, has points in six straight games.
"We were really focused tonight and we kept their skilled players at bay," said Tavares, who led the Islanders with 28 goals last season. "I like the way we kept it together in the third. It was a good way for us to finish it off."
Okposo had scored his second during a power play at 4:54 of the middle period to make it 2-2. Josh Bailey put New York ahead 1-0 in the first before Hall struck.
Hall gave the Oilers a 2-1 lead when he scored a pair of goals just 8 seconds apart in the first. That broke the club mark of 9 seconds, set by Wayne Gretzky on Feb. 18, 1981.
"That is so cool, good for him, he's a great kid. Records are so made to be broken, it's good for the game," Gretzky told ESPN.com's Pierre LeBrun via text message. "If Kevin Lowe would have got me the puck quicker it would have been seven seconds!"
The spurt was little solace for Hall and the struggling Oilers, whose only win was a 5-4 shootout home victory over New Jersey on Oct. 7.
"It's an empty feeling because a loss is a loss," said the 21-year-old Hall, the No. 1 pick in the 2010 NHL draft, one year after Tavares had the same honor. "It's great to know I have the record but tonight wasn't our best effort. The wheels have really fallen off for us as these losses have piled up."
Hall, who has three goals on the season, displayed plenty of frustration.
"We had the lead and chance to grab the reins and we didn't do it," he said. "We should have done better. It's tough."
Edmonton has had its struggles on Long Island, too. This was the Oilers' seventh straight road loss to the Islanders and their ninth in 12 overall meetings with New York.
They haven't beaten the Islanders on the road since Dec. 14, 1999.
The Islanders, 2-0-2 at home, surrendered late tying goals to Columbus and Buffalo before losing in shootouts at Nassau Coliseum. Against the Oilers, they played a strong third, outshooting Edmonton 15-8 and giving the Oilers few quality chances against Evgeni Nabokov, who made 27 saves.
Okposo's power-play goal, which came while Luke Gazdic served a tripping penalty, pulled the Islanders even.
After Bailey opened the scoring at 9:59 of the first, Hall tied it with at 15:52 with a wrist shot from the left circle. He then knocked a rebound past Nabokov at 16:00.
Except for the lapse during Hall's scoring spurt, the Islanders dominated a briskly played first period, outshooting the Oilers 16-7. New York outshot Edmonton 40-29 overall.
New Oilers coach Dallas Eakins was showing signs of exasperation, usually reserved for midseason slumps when teams tumble from playoff contention. After only eight games behind the bench, he was shaking his head.
"I knew this renovation was going to be messy, but I didn't think it would be this messy," he said of the Oilers, who haven't reached the playoffs since 2006.
The 38-year-old Nabokov made several sharp saves for the Islanders, particularly on a one-timer by Nail Yakupov midway through the second. He recorded his 338th career win, 18th on the NHL list.
"Your goalie has to be your best player sometimes, and Nabby certainly was, especially for the three penalties we had to kill," Islanders coach Jack Capuano said. "He was our best guy tonight."
The Oilers, who have allowed a league-high 35 goals, were coming off a 3-2 loss at Pittsburgh on Tuesday after a 4-2 defeat at Washington the previous night.
The teams will meet March 6 in Edmonton. ... The Oilers are 0-3-1 on a six-game trip that continues in Ottawa on Saturday and concludes in Montreal on Oct. 22. ... The Oilers were missing center Sam Gagner (broken jaw) and defenseman Corey Potter (back). ... Islanders fans booed Edmonton's Ryan Smyth every time he touched the puck. Smyth left the Islanders after playing only 18 games after a heralded trade late in the 2006-07 season.
Information from The Associated Press was used in this report.