Detroit, which broke the league's single-season record of 20 on Tuesday, surpassed the 22 wins in a row the Boston Bruins had over two seasons more than eight decades ago.
The Red Wings won their sixth straight game, all in a homestand at Joe Louis Arena, and have an NHL-high 84 points.
Detroit's Joey MacDonald, filling in for the injured Jimmy Howard, made 31 saves.
The Pacific Division-leading Sharks have lost six of eight.
San Jose's Antti Niemi gave up a goal on the second shot he faced and finished with 25 saves.
The Red Wings snapped a five-game losing streak against San Jose in the regular season, beating a team that has eliminated them in the second round in each of the past two postseasons.
MacDonald won his sixth straight game in place of Howard, who isn't being rushed to return after breaking his right index finger. Howard is expected to play Tuesday night at Chicago.
Detroit got off to a good start against the Sharks, getting goals from Zetterberg 6:30 into the game and Miller midway through the second, but it wasn't easy to beat them as usual.
Couture was credited for a goal after video review that bounced off the top of the net and appeared to be whacked into the net by Detroit's Justin Abdelkader on a power play with 5:27 left in the second period.
Miller's centering pass set up Helm for a shot from the slot in the third. Later, Marleau finished off a goal-mouth scramble by stuffing the puck past MacDonald.
NOTES: The NHL's second-longest winning streak at home this season is eight, shared by Calgary and Colorado. The second-longest active streak at home is five, shared by St. Louis and Phoenix. ... In the shootout's seven-season era, the previous mark for the longest home winning streak was 14 by Boston during the 2008-09 season. ... San Jose beat Detroit 5-2 in their previous meeting and won 4-2 at Joe Louis Arena on Oct. 28. . .... MacDonald shares the longest winning streak by a goalie born in Nova Scotia with Wendell Young and Pokey Reddick, who won six in a row for Pittsburgh and Winnipeg, respectively, two-plus decades ago. MacDonald won a combined six games the previous two seasons for Detroit and Toronto. ... San Jose D Jason Demers played for the first time in nearly a month. ... The Red Wings' next home game is Thursday night against Vancouver.