West Brom become the first team since Arsenal in September 2015 to leave City's fortress home with three points, in a run that spanned 20 games - the best anywhere in Europe's top five leagues.
Stand-in keeper Ron-Robert Zieler was called into action early on to deny Phillips, pushing his low shot from 15 yards out behind for a corner.
Solomon Rondon came close to making the breakthrough with a couple of headed attempts, but chances proved to be few and far between in a quiet opening 45 minutes at the home of the champions.
The game burst into life early in the second half when Morrison directed a header past Zieler from Phillips's deflected cross, but Leicester were level three minutes later as Islam Slimani directed Riyad Mahrez's cross past the reach of Ben Foster.
Phillips was needed up the other end soon after to clear Robert Huth's attempt off the line, and that would prove to be a big moment in the match as Albion soon pushed on to regain their lead.
Jamie Vardy and Andy King almost rescued a point for their side with near misses in the remainder of the match, but it was not to be for the faltering champions as the Baggies held on for just a second away win in 12 attempts.