India have won all three of their opening games, with the 29-year-old Bumrah taking eight wickets at a meagre average of just over 10 apiece.
But his career has been blighted by injury, with the unorthodox Bumrah sidelined for over a year following back surgery before returning in August.
"You have seen him in the last three games that he has played," Mhambrey told a pre-match press conference on Wednesday ahead of India's match against Bangladesh in Pune.
‘World class bowler’
"What he brings to the table -- he is a world class bowler. He gives you that breakthrough that you require in the power-plays.
"He's well adapted to bowl in the middle overs and he's a top-gun 'death' bowler. I think in that sense, we really missed him.”
Mhambrey added India had yet to consider "rotation" as they look to maintain their unbeaten start in a quest to win a third one-day international World Cup title and second on home soil following their 2011 triumph.
And that means the likes of proven performers in seamer Mohammed Shami and off-spinner Ravichandran Ashwin are set to remain on the sidelines.
Ashwin especially has often found himself omitted across all formats in recent times.
The 37-year-old was also left out of the team that lost to Australia in this year's World Test Championship final at The Oval.
"He (Ashwin) has been a great lad," said Mhambrey. "He understands that, he's a great team guy.
"I've never seen him grumpy; I haven't seen him complaining any time for any of the last few years that he's been with us.
‘After so many years’
Mhambrey added: "Even after so many years, I think he's there, he wants to do well for the team, he turns up every practice session, goes through the rigorous grinds and he keeps bowling.”
Mhambrey said the India bowlers that did play on Thursday would face a "challenging surface”.
"It (Pune) is a small ground, comparatively...It is at high altitude. So, I think here, more boundaries will be scored.”
Bangladesh have lost their last two matches at the World Cup after starting with a six-wicket win over Afghanistan.
They defeated India by six runs when the teams met in the Asia Cup in Colombo last month, with Mhambrey, highlighting the quality of Bangladesh fast bowler Taskin Ahmed and "champion" all-rounder Shakib Al Hasan
With Afghanistan having already pulled off a shock defeat of champions England and non-Test nation the Netherlands beating South Africa by 38 runs, Mhambrey said: "When you enter a World Cup, every team will pose you a challenge. So, from our perspective, I don't think we'll take anyone (lightly).
"Bangladesh or the Netherlands, I think we want to treat those games as equally important."