I have booked a 61 day trip to India but the E-TOURIST VISA is only 60 days. I arrive at 11:40 pm on the first day so wouldn't get through customs until the next day anyway, making it a 60 day trip. Would this be okay?
Regardless of whether you can do this, it's not a good idea. You've left yourself no flexibility at all to deal with problems at the end of your trip.
Flight home cancelled or significantly delayed? You've overstayed your visa. Late to the airport and missed your flight? You've overstayed your visa. On time to the airport but security was crazy that day and you missed your flight? You've overstayed your visa. Get sick and end up in hospital? You've overstayed your visa. Get sick enough to want to get over it before you fly? You can't because you'd be overstaying your visa.
Most countries are able to be accommodating if you have a genuine reason for overstaying but, of the ones I've suggested above, only "I was sick in hospital" is likely to be accepted. Anything else falls under "It's your own fault for trying to leave at the last minute, even though people miss flights and get sick all the time." (For example, here's what happens in the USA.) But even if you can get your stay extended due to some kind of emergency situation, this is likely to be a huge hassle.