The headline HSBC Services Business Activity Index was at 50.2 in May, rising from April's reading of 48.5 and pointing to the first expansion of output in 11 months.
A reading above 50 shows that the sector is expanding, while a reading below 50 shows that the output in the sector is contracting.
The rise was largely supported by a rebound in new orders and an overall improvement in client demand.
"Activity in the services sector is improving gradually. Fortunately, the boost to sentiment from the strong election result will further drive up activity in the sector," HSBC said.
The Indian economy will be able to tolerate only a gradual pick up in growth, otherwise inflation will rear its head again, HSBC said.
"PMIs indicate a gradual improvement in services activity, with the index creeping above the 50 level after nearly a year. The election result should add further strength to the recovery, but don't expect soaring growth," HSBC Co-head of Asian Economics Research Frederic Neumann said.
Earlier this week, the HSBC/Markit manufacturing PMI showed that the sector inched up in May driven by higher domestic and export orders.
Accordingly, the HSBC India Composite Output Index, which maps both services and manufacturing, posted 50.7 in May, up from 49.5 in April, indicated overall growth for the first time in three months.
On price rise, the report said, input costs faced by service providers in India continued to rise in May, and prices charged by services companies increased for a 43rd consecutive month in May.
On monetary policy going ahead, Neumann said: "The RBI is likely to retain its hawkish stance given the increased risks attached to inflation."
The RBI yesterday kept the key interest rate unchanged at 8 percent. However, it cut the amount of deposits banks need to park in government securities by 0.5 percent to 22.5 percent to improve availability of funds.
"The Reserve Bank remains committed to keeping the economy on a disinflationary course, taking CPI inflation to 8 percent by January 2015 and 6 percent by January 2016," RBI Governor Raghuram Rajan had said in the Second Bi-Monthly Monetary Policy Statement.