Richardson had five of New York's nine three-pointers and Chris Duhon tallied 19 points with five boards for the Knicks, who had dropped two straight and five of seven coming into Monday.
David Lee notched his 29th double-double of the year with 11 points and 10 rebounds. Only Orlando's Dwight Howard has more double-doubles this season, with 30.
Luol Deng and Derrick Rose each dropped 20 points with Deng collecting 10 boards and Rose handing out eight assists for the Bulls, who dropped their second consecutive game.
Tyrus Thomas finished with 19 points, 10 rebounds and three blocks while Kirk Hinrich had 13 points and seven assists in the setback.
The Bulls are just 5-16 away from the United Center this season and have lost four of the last six overall.
The Bulls briefly took a three-point lead as Andres Nocioni and Ben Gordon opened the final stanza with field goals, but New York regained the lead with a 9-0 surge.
A reverse layup from Rose and back-to-back buckets from Thomas, though, allowed Chicago to tie the game at 91 with just under five minutes left.
Rose then drilled back-to-back jumpers and Chicago went up three before Richardson buried a triple to tie the game at 97 with 2:15 left.
Hinrich hit just one of two in a trip to the line, allowing New York to go up one as Duhon drove the baseline and rolled it in for two with 31.2 on the clock.
Rose couldn't connect on a trip to the rim on Chicago's ensuing possession and Duhon pushed the lead to three with a free throw. Gordon's triple was off the mark for the Bulls and Nate Robinson sealed the win despite hitting just 1- of-2 at the line.
The Knicks held a slim lead, 22-20, following a Tim Thomas hook shot that closed out the opening quarter and took a 54-41 lead after Jared Jeffries capped a 9-0 spurt with a pair of free throws with three minutes to go in the first half.
The Bulls, though, scored the next eight points and ended the half on a 13-3 run, trailing by only three, 57-54, at the break.
Both Richardson and Al Harrington hit three-point shots to give New York a nine-point advantage with just under five minutes left in the third but Chicago again closed out the frame strong with a 13-5 run to get back within one, 78-77, heading to the fourth.