Rangers created a number of chances and Conor Washington’s deflected shot struck the City bar before Idrissa Sylla’s 75th-minute goal broke the deadlock.
The Guinea striker’s first-time finish from Massimo Luongo’s cross secured a first home league win since the opening day of the season.
“In the first half we were the better team and should have scored,” Hasselbaink said in his post-match press conference.
“It’s always difficult when you’ve played that well and are not in front at half-time, but we were persistent and did the same in the second half.
“We went for it. It was always going to be a battle in midfield and about winning the physical battle.
“Whoever won that could then create, and we did that particularly well.”
Rangers scored their goal three minutes after a crucial save from keeper Alex Smithies, who denied Bobby Reid at point-blank range.
“You need to be your goalkeeper and it was a great save,” said Hasselbaink.
“But then we picked it up and scored a really well-worked goal and I was pleased we could then hold on.
“My boys battled away. Yes we needed our keeper to make a good save, but we deserved the three points.
“It’s always important to win at home and obviously it was in the backs of our minds.
“We have been unfortunate in the last few games but we cannot look back. What’s important is what is now in front of us.”