Mark Noble had fired the visitors' into a first-half lead with a neat finish from ten yards.
But Andrews earned a valuable point for Sam Allardyce's side just minutes after coming off the bench.
El-Hadji Diouf was twice denied by the linesman's flag, as Rovers more than matched their loftier opponents.
There was mixed injury news for the two managers before the game, as defenders Ryan Nelsen and Gael Givet passed late fitness tests for the home side, whilw Matthew Upson and Savio Nsereko were ruled out for the visitors.
Allardyce had called for greater vocal support in the build-up to the game and, backed by an expectant Ewood Park crowd, Rovers made the better start of the two teams.
On eight minutes, Chris Samba flicked on Nelsen's ball into the box, but Benni McCarthy volleyed wildly over when Diouf was perhaps better placed.
Two minutes later, Morten Gamst Pedersen's corner caused chaos in the West Ham box and, when the ball eventually dropped back to the Norwegian winger, his shot was well saved by Robert Green.
The Hammers weren't without their chances too however. On 16 minutes, Noble fed David Di Michele, whose curling effort was comfortable for Paul Robinson and, just under ten minutes later, Rovers failed to clear their lines, but Diego Tristan was unable to force the ball home.
Rovers had the ball in the back of the net just past the half-hour mark, when Pedersen's long throw was flicked on by Samba and Diouf bundled the ball home, but the Senegal international was flagged offside.
At the other end, West Ham wasted a two-on-one, as Tristan was also flagged offside, but the visitors didn't have to wait long for the opening goal.
Noble started the move with a break down the right and, after Luis Boa Morte and Tristan had combined in the box, the ball was played back out to Noble, who stroked home his fifth of the season with the outside of his boot.
Allardyce brought on Andrews for Andre Ooijer at the break and the substitute made an immediate impact.
On 50 minutes, the Irishman's cross found Jason Roberts at the far post, but his crisp strike was blocked on its way to goal. From the very next play, Pedersen's long throw was only partially headed clear by Herita Ilunga and Andrews drilled the ball home through a crowd of bodies.
Pedersen volleyed well wide just past the hour mark and, two minutes later, Diouf again thought he'd opened his Ewood account.
Samba's left-foot strike was well kept out by Green and Diouf pounced on the rebound, but again the former Sunderland striker had strayed offside.
Rovers laid siege to the West Ham goal but, time after time, the visitors made vital blocks from strikes on target.
Gianfranco Zola brought on Kieron Dyer for Di Michele and the former England international thought he'd won a late penalty after a tangle of legs inside the box with Samba, but referee Chris Foy remained unmoved.