The Revolution (5-4-5), coming off a 5-0 rout of Los Angeles last week, dominated play in the second half and looked to go up in the 66th, when Fagundez launched a blast toward the net, but Hamid tipped it over the crossbar.
Chris Tierney had another good chance for New England a minute later, but his shot banged off the post.
Hamid also made a first-half save on New England's Juan Toja in the first half.
United (1-10-3) extended its club-record winless run to 12 matches.
Notable deaths of 2014 [PHOTOS]