David Mabey and Charles Forsyth were found guilty at Southwark Crown Court of the offences in 2001/2.
The company, which admitted breaching UN sanctions, was fined £5m in 2009.
Mabey, 49, from Burchetts Green, near Maidenhead, Berkshire, and Forsyth, 62, from Hook, Hampshire, will be sentenced on 23 February.
A third defendant, former sales manager Richard Gledhill, 64, from Watford, Hertfordshire, pleaded guilty and gave evidence for the prosecution.
Mabey & Johnson, which is part of theMabey Group, won a contract under the UN Oil For Food Programme, which was designed to alleviate the suffering of the Iraqi people.
But the court heard the company made a dodgy payment of £35,5000 in order to secure the £3.6m contract to supply 13 steel modular bridges.
Gledhill told the court he negotiated the contract with the Iraqi government and obtained approval from Mabey and Forsyth to make the payments via an agent.
The kickback was disguised as “commission” payable to Mabey & Johnson Ltd’s local representative, Upper Gulf Agencies (“UGA”) and was factored into the contract price by increasing the price.
Mabey’s solicitor, Stephen Gentle, said: “My client is surprised and profoundly disappointed by the jury’s verdict today.
“He maintains he was personally unaware of any breaches of UN sanctions against Iraq within his company in 2000-01. He has instructed me to seek leave to appeal.”
A spokesman for Mabey Bridge Limited said: “We recognise that the jury has heard the evidence and has taken its decision.
“As a company, we have already dealt with the issues in this trial, notably by pleading guilty to wrongdoing in a separate trial in 2009 and by paying a substantial financial penalty.
“We took responsibility for these issues, co-operated fully with the authorities and completed a profound renewal of our company and its ethical procedures and processes.”