Winger Adomah struck in the 16th minute and substitute Clarkson scored in the closing stages to wrap up the Robins' first win of the season and lift them out of the Championship relegation zone.
Whether or not it was Nigel Adkins' final game at the Glanford Park helm remains to be seen as he has held talks with Southampton over the vacant managerial position at St Mary's.
The Iron were unchanged from the side which beat Crystal Palace on home soil a fortnight earlier.
Midfielder Cole Skuse was recalled to City's starting line-up after a knee injury and replaced Ivan Sproule on the left-hand side of midfield.
Scunthorpe began the brighter and could have led inside the sixth minute when Jonathan Forte crossed dangerously and Sam Togwell connected with a superb 25-yard volley which David James tipped around a post.
In the 15th minute, the Iron went close again as midfielder Michael O'Connor curled a delightful shot goalwards from 25 yards but the ball narrowly missed the target.
Yet two minutes later City led when Brett Pitman collected possession and provided a fine pass to Adomah, who rifled a right-footed shot into the bottom corner from 12 yards.
Scunthorpe were dealt a blow soon after when defender Rob Jones succumbed to injury and was replaced by Niall Canavan.
In the 25th minute, Eddie Nolan crossed dangerously but Damion Stewart produced a fine clearance.
Moments later Jon Stead hit a fine shot from 30 yards which David Mirfin did well to block.
Yet the Iron continued to probe and were almost rewarded in the 33rd minute when Collins had a fine long-range effort saved by James.
Two minutes later, Pitman fired wide when he should have done better.
Stead then had another effort blocked by Mirfin and Marvin Elliott headed a cross from Nicky Hunt straight at home goalkeeper Joe Murphy.
Four minutes after the restart, Chris Dagnall mishit a long-range shot and team-mate Josh Wright fired over from 25 yards.
In the 54th minute, Sproule replaced Adomah and Collins then had an effort saved by James.
Three minutes later, Wright crossed dangerously and Cliff Byrne connected sweetly only for his effort to be blocked by Liam Fontaine.
In the 62nd minute, Stead crossed for Pitman to fire straight at Murphy from 12 yards.
Chances continued to come and go but Scunthorpe could not find a way through as City held firm.
They claimed their second goal in the fifth minute of added time when he collected a pass from Stead and rifled home from 12 yards.