Sub-inspector M.L. Nijalingappa, 58, and constable H. Vishwanath Rao, 30, will be rewarded by their bosses for what tweeters have lauded as a courageous act in an age of "VIP culture".
The two policemen had stopped all traffic at Trinity Circle, a busy five-road junction, for the President's convoy to pass when, around 4.15pm, they briefly heard the wails of an ambulance siren.
"We didn't think twice before creating an opening for it," Nijalingappa, two years from retirement, told The Telegraph.
He added: "We just did our duty. There's nothing more to it than allowing a patient to be rushed to hospital."
Mukherjee, who was visiting the city, was not held up: his convoy arrived about 30 seconds after the ambulance had passed.
"Traffic PSI M.L. Nijalingappa, salute you sir for choosing humanity over hierarchy and the display of dedication," Vijay Shenoy, a Twitter user, posted.
Sabyasachi Das tweeted: "In the past patients have died waiting for VIP convoys. So, right behaviour needs to be recognised & rewarded. Glad common sense prevailed."
The two cops' act comes at a time Bangalore police have been under fire not just for the city's notorious, miles-long traffic snarls but also for their handling of VIPs.
On May 15, an ambulance carrying a heart patient had been stopped near the state Congress headquarters for home minister G. Parameshwar to pass. Just 10 days earlier, police commissioner Praveen Sood had directed traffic cops to allow all emergency vehicles to pass even if they had to stop VVIP convoys.
It was in this context that the Bangalore police publicised Saturday's act by Nijalingappa and Rao via Twitter.
"Almost every day I hear on my walkie-talkie that some traffic policeman has helped move an ambulance ahead of the convoys of the chief minister, home minister or the governor. But we never gloat about it," R. Hithendra, additional commissioner of police (traffic), told this newspaper.
"But this time we decided to tweet since the convoy was carrying our President. We just wanted to state that they (the two cops) had done their duty. That's what they are trained for, and that's what we have instructed them to do.... Still, we have decided to reward them."
Hithendra said Nijalingappa and Rao would receive Rs 2,000 in cash and a certificate besides positive entries to their career records.
Rao, who has been Nijalingappa's assistant for four years at the Ulsoor traffic police station, said he had just followed his senior partner's orders.
"We facilitate the passage of so many ambulances and other emergency service vehicles almost every day. On Saturday, we had just seconds left before the President's convoy was to arrive; so we had to hurry," he said.
Rao said around 200 ambulances passed the junction, located in the city's central business district, every day. "We were lucky that only one ambulance was there at that moment."