The League One side controlled the first half at Victoria Park and a 12th-minute goal from skipper Carl Baker looked like it might have been the first of many.
But Pools looked a different side after the break and deservedly earned their place in the third round draw after Andy Monkhouse headed in an equaliser after 78 minutes.
"Coventry are one of the best footballing sides in League One and we've given them a game," said Cooper.
"I said during the week we would. Well we did in the second half, maybe not in the first half.
"We got on the front foot in the second half and pressed them and put them under pressure in and around their half of the field rather than giving them comfortable possession.
"They got into our half too easily in the first half but credit to the lads, they took on board what we said at half-time and they really stepped up their game in the second half and were a real credit. I was really proud of them.
"We made some changes and the subs made an impact and that's what they're supposed to do."
- Sports & Recreation