Matt Mills' header cancelled out Kurt Zouma's debut goal for Chelsea before Oscar netted a second-half winner to set-up a fourth-round tie at League Two Shrewsbury at the end of October.
"We were very happy with the performance from the team tonight," said Holland, undertaking media duties in place of boss Jose Mourinho.
"It's nice not to have extra-time
We don't have any new injuries, we've progressed to the next round of the competition.
"Several of our players who probably needed a game now have 90 minutes under their belts heading into two or three important fixtures
It's a good situation to be in."
Mourinho made nine changes to the side which began Sunday's draw at Manchester City, giving his fringe players the chance to impress.
Goalkeeper Petr Cech was captain, starting for the first time this season after being displaced as first choice by Thibaut Courtois, but his services were barely required as the Blues dominated.
Andre Schurrle spurned a host of chances and Bolton goalkeeper Andrew Lonergan was in fine form to deny the hosts.
World Cup winner Schurrle continually tried and failed to find the net, coming closest in the second half when his shot was tipped on to the post by Lonergan before ricocheting back into the goalkeeper's grateful arms.
"Andre's performance reflected the mentality of the team," Holland added.
"When you think not much more than two months ago he was playing in a World Cup final, tonight he's playing in the third round of the Capital One Cup and showed a really committed performance, full of ambition and drive all night, was dangerous throughout the game, looked our most likely source."
The profligacy of the German and Chelsea must be a concern, though, so too the defensive set-piece which saw nine-club journeyman Mills head in an equaliser to deny Cech a clean sheet and leave the normally miserly Blues with just one shut-out in seven matches this season.
Holland said: "I thought it was probably their only way that they looked dangerous all night, the set-plays.
"It was a concern for us all night, the set-plays, which really was a reason for the late substitutions (bringing on Didier Drogba and Nemanja Matic), as much to make the team bigger just so that we reduced the risk of that late killer goal from a corner that would have taken the game to extra-time.
"Had that happened that would have been very unfair on the team
They didn't deserve that."
Oscar saved his side's blushes by striking from 25 yards out after 55 minutes for what proved to be the winner in a contest which should have been far more comfortable for Mourinho's men.
"It was a real quality strike," Holland said.
"We said at half-time that we were happy with everything but it needed one or two individuals just to produce that little bit more quality to make the difference.
"That moment of quality has made the difference
It's won the game."
Bolton have one win in eight Championship matches this term but tested Chelsea.
Wanderers boss Dougie Freedman said: "I felt when I saw the team-sheet that I knew their manager treated this competition with respect and the opposition with respect
They were taking it very seriously.
"We were going to have to ride our luck a little bit tonight
Our goalkeeper was fantastic
They hit the bar
We had a plan and the plan was to get into the second half still in the game.
"That's what happened
In the last 10, 15 minutes we gave it a right good go, played a bit more expansive.
"A little bit more quality would have taken it to extra-time.
"Overall I think we gave a good performance and our fans were excellent
We showed that commitment for them."
Source : PA
Source: PA