Accies suffered three successive defeats to start their season but they brought this run to a shuddering halt with a victory over their local rivals to haul themselves off the foot of the Premiership table.
Odoffin, a new recruit from Livingston, bagged his first goal for the club as he powered home an 86th-minute winner.
Motherwell are still striving for their first league win of the new season as they were unable to break down Hamilton's resolute defence, with goalkeeper Ryan Fulton pulling off some impressive saves from David Turnbull.
Hamilton were first to threaten after 12 minutes when Marios Ogkmpoe outjumped Motherwell goalkeeper Trevor Carson as they contested a lofted Declan Gallagher back pass and nodded towards goal. The striker, however, was unable to get enough power on his header as Gallagher got back to clear the danger.
The visitors then had a chance on the counter attack when David Templeton released Justin Johnson but the winger was hesitant as he tried to get past Allan Campbell and the opportunity was snuffed out.
Motherwell began to pose more of an attacking threat as the first half wore on with Mark O'Hara seeing a drive deflect off Odoffin and spin just wide of the post.
Then just before the half-hour mark, Turnbull drilled in a low shot from 25 yards that was heading for the bottom corner but Fulton managed to turn it round the post.
Two minutes before the break, Fulton pulled off an even more impressive save as he once again denied Turnbull.
The Motherwell midfielder got on the end of Ricki Lamie's cross and his glancing header looked destined for the back of the net until Fulton got a fingertip to it to push it behind.
Six minutes after the interval, Fulton denied Turnbull for a third time as the midfielder once again tried his luck from range as he let fly from 25 yards with a powerful drive but the keeper was well positioned to tip it over the crossbar.
Hamilton then had a chance when Motherwell defender Stephen O'Donnell failed to deal with a Lewis Smith cross and the ball broke to Ross Callachan but the midfielder just could not sort his feet out in time as Carson came off his line to block.
Accies maintained the pressure with a some teasing crosses as substitute Callum Smith came within inches away of turning one in at the far post.
Hamilton should have taken the lead when Reegan Mimnaugh's driven corner flashed across the face of goal but Odoffin was unable to apply the finishing touch from a couple of yards out.
The defender, however, made it up for that miss as he notched the opener with just four minutes remaining.
Once again the threat came from a Mimnaugh corner, which the defender met with a powerful header at the near post and beyond Carson to seal the three points.