Monk's former club Middlesbrough ruined both his birthday and his first game in the Blues dugout.
Monk said: "It was a tight game and we were unfortunate with the goal. We dominated the second half but couldn't quite get the equaliser.
"I was very happy with the commitment, desire and passion displayed by the players.
"There are things to be worked on in terms what has been happening over the last two years. No one could dispute the desire of the players to do well in the game."
Monk's hopes of a winning start disappeared when Bamford notched a 39th-minute goal to lift Boro into the play-off positions.
It also left their hosts in the bottom three but Monk said: "I have a group of players who can improve and from what I have seen I have the belief in our last 10 games we can achieve what we have to achieve.
"There is a lot of encouragement from the performance against Middlesbrough and the team showed they are going to fight to the end. It is all about building the confidence in the team.
"We limited a very good and potent side as we controlled long spells. The talent is in the side but it is also about getting points, which are so vital at this stage of the season."
Middlesbrough boss Tony Pulis was delighted with the win and said: "I never thought there was a time in the game they would score.
"I was pleased with the resilience of the lads as it was a difficult game for everyone.
"It was difficult with a new manager taking over Birmingham who was the last manager of Middlesbrough before I stepped through the door.
"You usually get the bounce effect with a new manager so it was always going to be tough for us. It was a great result for Middlesbrough.
"Going into the game we had scored nine goals in three games and we had been very potent going forward. I thought we had a lot of opportunities and I thought Stewart Downing could have scored a hat-trick.
"There are 10 games to go and we must not take the foot off the pedal as there are no easy games in the Championship."
Pulis paid credit to match-winner Bamford, saying: "Patrick is a good player and he has real quality."
The rejuvenated Patrick Bamford continued his scorching run of form with his seventh goal in the last four games to seal victory for Middlesbrough.
It meant Birmingham’s new manager Garry Monk – former Boro boss – was shown the size of the task facing him as the Blues’ suffered their sixth successive Championship defeat in his first game in charge.
It was an unhappy 39th birthday for Monk as his successor Tony Pulis, making his first return to the Midlands since being sacked by West Brom, chalked up his seventh win in 14 games since taking over at Boro.
Birmingham made a positive start, but it was Middlesbrough who appeared to be the more dangerous. Bamford displayed his potential threat when he set up Mo Besic, but he saw his close-range effort deflected for a corner.
Boro were playing with cool assurance and applied strong pressure, with Adama Traore in particular proving troublesome with his trickery and pace.
Birmingham’s response was a powerful long-range effort from Craig Gardner that had former Blues goalkeeper Darren Randolph scrambling to his right to save, and the reinstated Jota sent an effort into the side netting.
One pleasing feature for Birmingham was the return to form of Jacques Maghoma. Given the freedom to roam by Monk he was revitalised compared to recent lethargic form.
Resolute defensive play kept Birmingham in the game, but they finally succumbed in the 39th minute when a left-wing cross from Traore went through a packed defence and found Bamford on the far post, where he turned home the ball from close range.
Birmingham strived to get back into the game, but they highlighted why they are the team with the worst goalscoring record in the entire Football League.
Jota continued to be a potential threat before the introduction of striker Sam Gallagher and Jeremie Boga failed to produce the elusive equaliser and the hosts remain in the relegation zone.
Birmingham City: Stockdale, Jenkinson, Morrison, Dean, Colin, Gardner, Kieftenbeld, Jota, Adams (Boga 75), Maghoma, Jutkiewicz (Gallagher 60). Unused substiutes: Roberts, Bramall, N’Doye, Lowe, Kuszczak.
Middlesbrough: Randolph, Shotton (Cranie 71), Ayala, Gibson, Friend, Clayton, Besic, Traore (Baker 83), Howson, Downing, Bamford (Assombalonga 77). Unused substiutes: Konstantopoulos, Fry, Johnson, Harrison.
Referee: Stephen Martin (Staffordshire).