Borough held their three-divisions-higher opponents at 0-0 for nearly an hour before two goals from Gary Madine and one from Jerry Yates got the Tangerines through to the second round.
The Sports can be proud of their efforts - and must use the experience to help them build on a good start to their National League South season.
Bloor told BT Sport after an absorbing live televised tie: "I thought our boys were outstanding - 0-0 at half-time, that's all you ask for - to be in the game.
"We had a great chance early in the first half. Greg Luer perhaps would have done better on a normal day. But I couldn't be any prouder of my football club, my players and the whole occasion.
"Many congratulations to Blackpool - I felt Gary Madine was the difference between the two sides, a former Premier League side. But I'm absolutely delighted and proud of my club. Now we look forward to our league campaign.
"I felt our boys were great. We showed good energy, good enthusiasm. I felt our centre halves, big Mitch Dickenson and Steven James, did really well and I thought James Ferry was excellent in the middle of the park but we're out.
"Congratulations to Neil Critchley, Colin Calderwood and the Blackpool team, I'll be rooting for them for the rest of the season and in the next round. And we look forward to continuing our good start in National League South."