The two goals, 28 seconds apart, marked the second time Anaheim has scored twice within a 30-second span this season (also Nov. 6 vs. FLA).
With two assists on the night, Zegras is now one point shy of 100 for his 130-game NHL career. Kariya (98) and Bobby Ryan (126) are the only Ducks to accomplish the feat in less than 140 games.
Klingberg, who would add the decisive goal later on, has 3-7=10 points in his first 24 games as a Duck and eight points in his last 15 appearances.
That sudden two-goal lead would last all the way until the early minutes of the third, when Caufield followed up his longtime buddy Zegras' quick-strike first period with a pair of goals himself, scored 3:26 apart.
The first came on a power-play one-timer from the left circle, beating Dostal just inside the short side post despite a good read from the Anaheim netminder.
Jonathan Drouin grabbed the secondary assist, his first point since returning to the lineup after a month-long absence due to an upper-body injury.
Former third-overall pick Kirby Dach also grabbed an assist and now has points in back-to-back games. Dach ranks fourth in career scoring among players selected in the 2019 NHL Draft, trailing only Zegras, New Jersey's Jack Hughes and Buffalo's Dylan Cozens.
Caufield scored again on his next shift, tracking down an errant shot behind the net and winning the race to the far post on the wraparound to even the score at two.
The 21-year-old Caufield, who played his final pre-draft seasons with Zegras at the U.S. National Development Team and ranks fifth among those 2019 draftees, has goals in four of his last six games and a team-leading 19 on the season, tied for seventh among league leaders.
Just minutes later, the newest Duck, Jayson Megna, came mere inches from answering right back for Anaheim on an odd-man rush, beating Allen over the shoulder but ringing it right off the left post. Shattenkirk did the same on the next shift, getting his shot through traffic from the right point but finding nothing but iron as a reward.
But for visitors it would be the third time that proved to be the charm, with the Ducks finally breaking through for the go-ahead goal on Klingberg's second of the night. Vatrano worked the puck off the left wing wall, spotting Klingberg open at the far point. The first-year Duck defenseman took the pass from his fellow offseason signee, quickly winding up to use Canadiens defenseman Jordan Harris as a screen and firing his shot past Allen to the blocker side.