The Saddlers started slowly in the first half and struggled to create going forward, but a tactical switch at half-time gave them more attacking intent and after 70 minutes Osadebe smashed a wonderful effort in off the underside of the bar.
Jay Spearing picked up a second yellow card minutes later as Walsall comfortably held on to their lead, giving them two consecutive victories and the double over Tranmere this season.
Brian Dutton made one change to his side, starting in a 5-2-3 formation.
Tom Leak came in for the injured Hayden White, who was not involved in the squad. Caolan Lavery then returned from his injury and took the space available on the bench.
Callum Cockerill-Mollett and Jack Nolan were not involved, while Dan Scarr and Stuart Sinclair missed out through injury.
Keith Hill made two changes to his side, sticking with their 4-1-2-1-2 formation.
Veteran striker David Nugent came in, as Danny Lloyd dropped to the bench.
Spearing recovered in time from his injury to start at the base of the diamond in midfield, meaning Lee O’Connor was named among the substitutes.
Striker James Vaughan, who has been unavailable since late February through injury, returned to the squad for the first time and made the bench.
Tranmere dominated the early stages and almost took the lead within five minutes. A corner found Kaiyne Woolery who had a free header from six yards, but the striker totally missed the ball and the chance fell by the wayside.
The striker recovered quickly, however, when minutes later he danced beyond several Walsall men and got a shot away from 20 yards, which was deflected wide. Peter Clarke met the resulting corner but his header was straight at Jack Rose.
Walsall grew into the game, however, and were using their wing-backs and midfield to press Tranmere high up the pitch and win the ball back. Josh Gordon did well when he robbed Ali Crawford and took aim at goal from 25 yards. His strong effort was palmed away by Joe Murphy.
Liam Feeney should have done better when he expertly brought the ball down and skipped beyond Leak, but then directed his effort high and wide of the Walsall goal.
Crawford was then played into space in the box and with time to make his move he smashed an effort into the side netting.
It was a heart in mouth moment for the Saddlers when Rory Holden went down holding his knee, having only recently returned from a serious knee injury. After Lavery got ready to replace him, Holden was eventually able to continue and stayed on the field.
After a fairly entertaining start to the game, in the first 10 minutes, the game dipped into a lull for the remainder of the half. By the 40-minute mark neither side had an abundance of chances and while Tranmere looked after the ball well, Walsall struggled to get a foothold in the game.
The teams entered half-time drawing 0-0 with Walsall failing to threaten. Holden was kept quiet and, in turn, so were the Saddlers.
At half-time Walsall brought on Cameron Norman, replacing Leak, and changed to a 4-3-3 formation. Osadebe moved into midfield, Holden played on the right and Tyreik Wright moved to the left.
Osadebe won a free-kick 20 yards out when Spearing flew into an awful challenge after the midfielder broke away and raced towards goal. He was booked for his troubles and Norman sent the set-piece over the bar.
Walsall’s best chance of the game came minutes later when Holden picked up the ball on the left following a deep cross that evaded his team-mates. He lifted the ball back in and Gordon headed towards goal, only to see Murphy leap to meet it and tip it just over the bar.
Gordon then headed just wide from a Norman cross, as Walsall looked far brighter in the second half after changing their system.
In an effort to find that first goal Wes McDonald and Lavery were brought on for Holden and Gordon, with 20 minutes plus stoppages left to play.
After 70 minutes that goal finally came. Osadebe picked up the ball centrally, cut inside one Tranmere player and then smashed a right-footed shot in off the underside of the bar from 25 yards, for his first goal from open play in a Walsall shirt.
To make matters worse for Tranmere, Spearing was given a second yellow card for a late challenge on Liam Kinsella, reducing his side to 10 men.
Walsall were defending their lead well but deep into six minutes of added time the visitors were appealing for handball on Mat Sadler and a penalty. Referee Ollie Yates was unmoved.
In the end, the Saddlers hung on for the three points to earn back-to-back victories for the first time since mid-December.
With this fixture the game in hand over Southend, it moved Walsall 12 points clear of the bottom two with only five games left to play.
Walsall: Rose, Leak (Norman, 45), Clarke, Sadler, Melbourne, Kinsella, Perry, Osadebe, Wright (Bates, 93), Holden (McDonald, 68), Gordon (Lavery, 68).
Subs not used: Roberts, Vincent, Osei Yaw.
Tranmere: Murphy, Khan, Clarke, Monthe, MacDonald, Spearing, Lewis, Crawford (Lloyd, 62), Feeney (Morris, 80), Woolery (O’Connor, 62), Nugent (Vaughan, 68)
Subs not used: Jones, Ridehalgh, Ray.