It was a far from vintage display by the Premiership champions who may well have had one eye on Tuesday's Champions League showdown in Barcelona.
But it was good enough to give Mourinho victory number 79 from his 100 games in charge.
Chelsea only came to life in the second half with Drogba setting up the success in the 50th minute with his 11th goal of the season and his third in the last three games against Albion.
Chelsea had to play the final 30 minutes with ten men after Arjen Robben was dismissed by referee Mark Halsey for a foul on Jonathan Greening.
But that proved to be no handicap and substitute Cole put the result beyond doubt with his ninth goal of the season in the 74th minute.
Albion did grab a consolation two minutes from time when Kanu turned home Paul Robinson's cross at the near post but Bryan Robson's side blew their chance to pull off a shock win in the first half when Chelsea really were there for the taking after a rocky 45-minute display.
Albion were dominant at that stage but could not turn their possession into goals - a costly problem that has only served to increase their relegation problems following their third successive defeat.
Robson chose to start with Kevin Campbell as a lone striker and a five-man midfield. It was a surprise tactic that Chelsea struggled to get to grips with until the second half.
They needed a smart ninth minute save from Peter Cech to keep them on level terms when Campbell headed a Diomansy Kamara cross goalwards.
When Jan Kozak lifted a volley over the bar four minutes later after Cech punched clear a cross from Greening, Albion could consider themselves unlucky not to be ahead.
Chelsea looked to be badly missing their injured leading scorer Frank Lampard at the heart of midfield and it was fully 30 minutes before they forced Albion goalkeeper Tomasz Kuszczak into a save and then he was able to deal comfortably with a weak shot from Damien Duff.
Albion continued to look the more menacing and they were inches away from making their pressure pay in the 41st minute when Kamara was released by Ronnie Wallwork after Junichi Inamoto had dispossessed Robben.
Kamara raced 30 yards before slamming his low shot into the side-netting. It proved to be a costly miss as a half-time pep talk by Mourinho paid dividends with Duff robbing Inamoto and setting up Drogba who drilled a low left-foot shot past Kuszczak.
When Robben saw red, Albion sensed a chance to get back into the game. But Cole snuffed out that hope 11 minutes after his introduction.
Cole, England's hero in midweek against Uruguay, cashed in on the hard work of Drogba who saw his shot crash off Wallwork and into the path of Cole whose close-range drive bounced over Kuszczak.
Kanu's late strike gave Chelsea a nervy finale but they stood firm to move closer to retaining their title.