Boro deserved the lead and looked in complete control once Ikpeazu, who moved from Wycombe this summer, had found the net with seven minutes remaining of the first half.
But Nigel Pearson’s Leicester old-boys, Matty James and Andy King, combined to level through the latter’s header on the hour.
The impressive Crooks – an arrival from Rotherham United – was then on hand 10 minutes later to grab the winner for the home side from inside the area.
That made it the perfect outcome for Boro boss Neil Warnock who, despite being in charge for more than a year, was overseeing his first Middlesbrough game in front of a big Riverside crowd.
It looked like Bristol City would make it hard for the hosts after a decent opening five minutes from the visitors.
But from that moment on, the Robins struggled to make an impact in the final third until the equaliser and it was the home side that created the first half-chances.
Ikpeazu was lively enough along with Crooks, Isaiah Jones and Djed Spence in the final third for Middlesbrough.
Jones – after impressing as a substitute at Fulham six days earlier – was quick to make his mark again with a run and pass into the area for Crooks’ first time effort to require a save from goalkeeper Dan Bentley.
A couple of minutes later, Crooks was involved again. This time he got to the byline to cross into the danger area where Spence arrived to head the ball down and wide.
And when Ikpeazu had dragged an effort wide following a nice pass from Paddy McNair, there was a sense that Bristol City were about to cave in.
In the 38th minute, Crooks broke into the visitors’ half and rolled Jones in down the left. The young winger beat Zak Vyner before his delivery into the box was only helped on by Rob Atkinson for Ikpeazu to hammer the ball in from close range.
Jones was quick to make an impact after the restart too. After cutting inside from the left again, his finish was not as impressive and he rolled his effort into the hands of Bentley.
But after Spence had hit the palms of Bentley following another Jones pass, City found a new lease of life and they were level on the hour from the Foxes combination.
James’ delivery from wide on the right into the area was watched by red shirts and that allowed King to coast in and head beyond goalkeeper Joe Lumley.
Middlesbrough regained the lead 10 minutes later as Ikpeazu’s knock down from McNair’s deep cross allowed Crooks to move ahead of his marker and finish.
There were a couple of late scares from the away side but Boro held on.