David N'Gog scored an outstanding opening goal before Tangerines forward Tom Barkhuizen's netted a controversial equaliser just before half-time.
But N'Gog then worked hard to set up Jermaine Beckford for the winner on the counter-attack in the second half as Bolton progressed through at the expense of their Sky Bet Championship rivals.
Freedman, whose side are in the bottom seven of the Championship standings with only six wins from 24 games this season, said: "I like a cup run. It helped us last year where we beat Sunderland and that got our season going.
"It was a good old fashioned cup tie today and I would rather be in the next round then not."
He added of his side's performance: "David was fantastic but I don't like singling out any individuals. I felt that our desire in attack was right up there.
"We could've defended some situations a little bit better but the real honesty and desire of the players to win was the most pleasing thing."
Freedman also stated his intentions in the transfer market, but did admit he may need to sell to bring players in.
"The four guys at the back have played most of our games and we've only really got those four defenders in the building so that's where we're looking to strengthen," he said.
"But we've got to cut our cloth accordingly and look for the right sort of deals. It's a fact, that's where we are right now and it's about getting the right deals in.
"I've got to make sure that I work really hard to get things right in the transfer market but the guys upstairs have been really good to me and they've always supported me."
Blackpool assistant manager Steve Thompson was relatively upbeat after the game despite defeat extending their winless run to eight games.
"We've got to take the positives out of the game," said Thompson. "We created a lot of chances and their keeper's made a lot of saves so it's disappointing that we didn't win the game when we played like that. On another day I think we come through.
"We looked like we had energy today and even in the defeats we've had we've shown what we can do.
"We just need that little bit of quality to finish that off which I think that we do have in the team."
Meanwhile, despite the Trotters complaints, Thompson believed the Tangerines' equalising goal should have been given.
Barkhuizen magnificently curled the ball past Andy Lonergan but celebrations were then cut short as the assistant referee had his flag up for interference in the build-up.
However, after consultation with referee Simon Hooper, the goal stood, to the disgust of the Wanderers players.
"It's one of those grey areas but they couldn't disallow that because it was a fantastic strike," Thompson said.
"I'd have been disappointed if we didn't get it, but likewise we were disappointed that we conceded that goal just after half-time."
Thompson also admitted that Blackpool were looking to recruit imminently, adding: "There's a lot of things ongoing at the moment and hopefully it will be a busy week for us."