It is their first league title since 2014 and their first under former Barcelona and Bayern Munich coach Pep Guardiola.
The match had looked on course to end in a stalmate, with top-drawer saves from David De Gea and Ben Foster the highlights before Rodriguez's goal from a corner handed a third Premier League crown to City.
The win was West Brom's first since a 2-0 success over Brighton on 12 January and only their fourth all season.
"It was a bad reaction after the City victory, the effect of that is complicated football", Mourinho said.
United have 71 also with five to play.
José Mourinho's United was at home to West Bromwich Albion, rooted to the foot of the table, all but assured of relegation, and coming to the end of a season in which it has fired two managers, a chairman, a chief executive and a director of football. They lost only two matches because they were the best team.
But Jay Rodriguez's goal in the 73rd minute gave West Brom a glimmer of hope as they are now nine points off safety with 12 to play for.More news: Indiana Pacers Playoff Preview: 4 Factors That Will Decide the Series
"We didn't want or played simple, everything was slow, one more flick, trick, turn".
There was enough in Sunday's performance to suggest caretaker manager Darren Moore should stay in charge.
While still nine points from safety, it was an incredible end to a match that had started with Livermore showing composure that belied his role as a midfield enforcer as he cut away from Victor Lindelof to be denied by David De Gea. "You have to be consistent, not just one wonder season", Giggs said.
Paul Pogba's two goals against City felt like a defining moment for the France midfielder during a season which has had more downs than ups.
Thierry Henry: "I am not surprised with what Pep has done in this league and a year ago, I was one of the people saying stop having a go at the man because that's how he plays".
Jay Rodriguez secured rock-bottom West Brom a win at Manchester United that was as remarkable as it had looked unlikely.
In the past five seasons no side has picked up more points at Old Trafford than West Brom, who have taken 10 over that period.
This was West Brom's first league away win in 16 attempts and ended a 12-match streak without a victory - a run which included 11 straight defeats. Eight titles. Three Premier League.