Motherwell's Michael Higdon cancelled out Nick Riley's opener for Dundee as the points were shared in a 1-1 Fir Park draw.
The rock-bottom Dark Blues, buoyed by their victory over Hearts last weekend, began the game confidently but Steven Milne shot tamely past Randolph's post in the second minute.
Milne and his Dundee team-mates then had a penalty claim rejected on the quarter-hour when the striker hit the deck under pressure from Micky Cummins.
The Steelmen took control of the game after that and Keith Lasley and Nicky Law both had shots pushed away by Douglas before Jamie Murphy fired just wide from 22 yards.
Motherwell came closer when Higdon headed Tom Hateley's corner past Douglas but Gary Irvine acrobatically cleared off the line.
Jim McAlister then set up a rare first-half foray from the visitors and involved Riley and Iain Davidson for Riley to sweep home the opener after Hateley blocked the latter's shot.
Well had failed to score in five of their previous six matches and they looked stunned to be behind as the visitors grew in confidence for Milne to shoot wide after Colin Nish had dummied McAlister's inside pass.
Both Higdon and Murphy were off target from Hateley crosses as the hosts searched for an elusive breakthrough and Well's Randolph got down well to hold Matt Lockwood's 25-yard free-kick.
Motherwell finally made their pressure pay though after Law's half-cleared cross fell to Ojamaa 20 yards out and the Estonian fired a powerful left-foot strike that Robert Douglas stopped only for Higdon to follow up and tap home (45).
Randolph saved the hosts after the restart when he reacted well to push the ball over after Simon Ramsden had got a head to Milne's overhead kick following a long throw.
Douglas then pushed away Murphy's drive as the hosts piled forward for a precious winner and Higdon had a shot well blocked after a corner.
The home fans were celebrating with three minutes left after Higdon volleyed home from inside the six-yard box following Lasley's shot spinning up in the air off a Dundee defender - but the offside flag was raised.