Peter Utaka claimed the only goal in the 28th minute with the Danish side's only real opportunity of the second leg - but Motherwell's failing came at the other end.
John Sutton and Chris Humphrey missed clear opportunities inside six minutes and the best of numerous second half opportunities came when Jamie Murphy missed a late penalty after Espen Ruud and Anders Moller Christensen had been sent off.
Motherwell had gone into the game full of optimism after looking at least the equal of the Danes in last week's 2-1 defeat, although they needed Tom Hateley's spectacular injury-time free-kick to give them and the vast majority of 9,105 fans true hope.
Nick Blackman retained his place after marking his first start with the only goal at Kilmarnock on Sunday. The on-loan Blackburn player replaced the suspended Steve Jennings in the only change to Motherwell's first-leg team.
Well's attacking line-up showed no nerves at the start of the biggest game at Fir Park for decades and Mohamed Traore survived handball claims in the opening minute.
Keith Lasley's shot was blocked, John Sutton's clever run ended with a shot into the sidenetting, then Humphrey sliced wide from eight yards following Stevie Hammell's cross.
Hateley was given another chance to shoot from just outside the box after Blackman won a foul, but the midfielder's low drive was well held by Roy Carroll.
Odense were not sitting back though and the return to fitness of Henrik Toft, whose burly physique and face mask gave him more the look of a wrestler than a footballer, allowed them to get the ball forward quickly.
The Danes looked dangerous on the break but vulnerable at the back and Ruud and Christensen were booked after illegally trying to atone for some shaky defending.
Motherwell got in behind the Danes again when Lasley's long ball sent Murphy into the left channel but he lobbed the ball just beyond the far post after Carroll had raced off his line.
Brown's side soon paid for their wastefulness. Ruud's inswinging corner was aimed towards Toft on the goal line, his physical presence prevented Well clearing properly and Utaka drove home from eight yards.
Instead of needing one goal to progress, Motherwell now needed two to force extra-time.
Hateley volleyed over from distance, Ruud looped a header on top of his goalkeeper's net and Blackman headed Lasley's cross over.
Several stoppages spoiled Motherwell's momentum but Carroll reacted well to hold Sutton's header following Murphy's cross.
The second half continued in the same manner with the hosts putting Odense under pressure but needing to be wary of counter-attacks.
Greek referee Michail Koukoulakis waved away penalty appeals when Sutton appeared to be held by a struggling Atle Roar Haland, who soon trudged off injured.
Carroll punched away Hateley's dangerous free-kick and the former Manchester United goalkeeper held Blackman's 20-yard drive.
Motherwell appeared to be in danger of losing their attacking drive but they squandered another glorious chance in the 70th minute.
Sutton headed Stephen Craigan's long throw straight to the unmarked Blackman but he headed weakly at Carroll from six yards.
Humphrey's cutback soon found Sutton 10 yards out but he could not position himself to shoot before being surrounded by defenders and his stabbed effort was blocked by Carroll.
Motherwell were handed a lifeline in the 76th minute when Ruud received a second yellow card for a needless foul on Blackman.
The chances continued to come - Murphy failed to connect with Sutton's inviting cross and Motherwell appealed strongly for a penalty as Chris Sorensen appeared to clear Hammell's cross with the aid of an arm.
Darren Randolph made his first save of the night in the 84th minute when he stopped Eric Djemba-Djemba's long-range strike and Motherwell's chances appeared to be vanishing.
But they were finally awarded a penalty a minute later when Christensen handled Hateley's cross as he challenged Sutton, the foul earning another yellow card.
Motherwell's failing over the two legs continued to haunt them though as Murphy, their leading European goalscorer, struck the post from the spot.