Oxigen USA website does the job with EXACT $ to ₹ exchange rate and zero fees. None. Zilch.
Accepted my EU debit card for an Indian mobile phone bill payment.
Small downside: my EU bank fee for converting € to $.
Fee can be avoided if you have a bank account with new age startup banks like Monzo (£), N26 (€) or Revolut (£/€).
Edit: now that the promotion period is over, the conversation rate is appalling on the website. Also the minimum transaction has to be Rs. 500. Still does the job though.