Some had booed Moyes during the game and questioned in song his decision not to use man of the match Khazri to any significant extent in recent months, although the Scot took it on the chin afterwards.
Crystal Palace recovered from two goals down to move seven points clear of the relegation zone.
On an eventful afternoon on Wearside, Andre Ayew fired the visitors ahead with just five minutes gone only for Khazri, starting his first game for the club since October 29, to level direct from a 26th-minute corner.
James Collins headed the Hammers back in front within two minutes of the restart, but substitute Borini struck in the 90th minute after Randolph's error.
"I don't think the crowd turned against the manager - not the way I've seen other managers treated", said the long-serving midfielder.
David Moyes praised Wahbi Khazri following his inspired display against West Ham United today.
Borini celebrated with gusto and Sunderland pushed for a victor in the 10 minutes of injury time, with Sam Byram sent off for a second yellow to heighten West Ham's anxiety, but that crucial third goal proved elusive. I have to accept it.
The signing of the Nigerian was met with bemusement initially but his hold-up play, not to mention his goals, gave Sunderland an important outlet.
Swansea City's survival hopes suffered another blow as Etienne Capoue's first-half goal condemned them to a 1-0 defeat against Watford at Vicarage Road. I just keep in mind to myself that I've got the third or fourth best win record in the Premier League.
Manager Slaven Bilic says his side "still have work to do" to make certain of their Premier League status for another year, although the Hammers are now closer to the top eight than they are the bottom three. I feel as though there are more things to worry about in terms of how we are at the moment as a team than my own situation. "I thought we were reasonably comfortable after we scored the second goal". That's happened a few times now but we're ready to prepare for Everton and that's going to be a tough game. In football there's an alternative opportunity and we've got another opportunity against West Ham.
Fans chanted for him to be sacked for the first time on Saturday, and this during one of their better performances in which they finally scored a goal after 701 minutes without one.