Town legend Danny Whitaker put the hosts in front before half-time but Harrison got a reaction from his players after the break, and Jake Cassidy equalised - watch match highlights here.
Following a home defeat to Dover Athletic on the opening day, a point at Macclesfield will have steadied the Pools' ship.
A disappointed first-half was followed up by a much better second, and Harrison told the club's website: "I thought the lads were brilliant in the second half and we finished the stronger, only one team was going to win it in the end. Obviously to go 1-0 down was very disappointing but it was good to see us show the fight, desire and the character from the players to come back and push to try and win the game.
“If you look at the bigger picture, a point away at Macclesfield who for the last four years have been in the top eight and competing for things at the top end of the table, we’ll be happy with that at the end of the season.
“The lads put a huge shift in and it was made even better by the travelling fans, who were fantastic.
“I think the fact that the players and staff went straight over to the fans at the end shows how much we all appreciate the support this evening and hopefully the fans all went home a lot happier this evening than they did on Saturday.”