The fourth seeded Indian pair dominated the quarterfinal contest, winning 6-4 6-4 in just 67 minutes.
One more victory will ensure India a silver medal and a defeat in the semifinals would give Sania and Bopanna a chance to fight for the bronze.
India has only one tennis medal in its entire Olympic history that came through Leander Paes, who won a singles bronze at 1996 Atlanta Games.
Murray came into the contest after winning his singles quarterfinal that lasted more than two hours and was a pale shadow of himself.
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Watson did not look like a player who won the Wimbledon mixed double title this year, struggling badly with her serve and ground strokes in the opening set.
Sania and Bopanna were a better team than the Britons, who never posed a threat to the Indians. Bopanna was solid with his booming serve while Sania played superbly from the back of the court.
However, the Indians immediately got the break back as Watosn dropped serve at love. A flurry of unforced errors by the Briton set Indians nicely with three chances and Bopanna found a backhand winner on the first break point.
Sania saved a breakpoint in the next and held to make it 2-2.
A nervous looking Watson struggled with her serve and was again down 0-40. The Indians seized the advantage. Murray failed to put across a strong forehand return from Sania, putting the Indians ahead 4-3.
The British players played better as a team in the second but the Indians were up for the task and broke Murray in the fifth for a 3-2 lead.
That lead stayed with the Indians and Bopanna sealed the issue on the third match point with an overhead smash.