While Red Bulls Designated Player Thierry Henry is one of the premier attractions in Major League Soccer, he did not make the trip to Foxborough, missing the match due to inflammation in his Achilles.
Caraglio got New England off the mark in the 15th minute when he collected a rebound at the top of the box and curled a shot just inside Bouna Coundoul's near post.
The Argentine doubled New England's in the 37th minute, nodding home a set piece with a brilliant snap-header for his second MLS goal.
Caraglio had a chance to nab a first-half hat trick in the 40th minute, but his close range effort was denied by a one-handed save from Coundoul.
Controversy struck just before halftime when New England goalkeeper Bobby Shuttleworth appeared to bring down Dax McCarty in the box, but the referee issued a yellow card to the New York midfielder for diving.
The Red Bulls were handed a couple lifelines in quick succession early in the second half.
Dane Richards pulled a goal back for New York in the 53rd minute, and Kenny Mansally was shown a red card in the 55th minute.
But the Red Bulls were unable to capitalize on the short-lived man advantage, as Teemu Tainio was sent off in the 69th minute after receiving his second yellow card.
Richards and the Red Bulls left it late though, with the Jamaican international converting a low cross to equalize in the 87th minute.
New York (6-6-14) has not recorded a win since defeating Toronto FC 5-0 on June 6. Since then, it has posted an 0-3-4 record an fallen to fifth in the Eastern Conference standings with 32 points on the season.
New England (4-11-11) is now level with Toronto for seventh place in the East with 23 points. Toronto will have a chance to break the deadlock when it faces the Chicago Fire on Sunday.
The Red Bulls will hope to regroup and get back into the win column when they host the Los Angeles Galaxy on Aug. 28. The Revolution will travel to PPL Park to face the Philadelphia Union the same day.