Alexander's strikepartner Jason Price - on loan from Doncaster - gave the Lions the lead 17 minutes in before a neat finish from Jabo Ibehre pulled the visitors level on the half-hour mark.
But the Lions' top scorer took his side into the break with their noses in front just a minute before the interval and then secured all three vital points on 52 minutes.
Despite an array of goalmouth action at both ends, the game was played at a sluggish pace similar to a pre-season friendly despite the win keeping Kenny Jackett's side in touch of second-place Peterborough - who visit them on Easter Monday.
Normally composed defender Zac Whitbread almost allowed Walsall in with his early mistake but the Liverpudlian covered to deny the lively Troy Deeney.
Price found himself with a great opportunity to put the Lions ahead three minutes before he eventually did as he forced a smart save from Rene Gilmartin in a one-on-one situation.
But the Doncaster outcast got his goal after tapping home Whitbread's brilliant knockdown from David Martin's hanging cross.
With the first half petering out, the guests struck against the run of play with a well-taken strike.
Former Leyton Orient striker Ibehre latched on to a loose ball in the Lions box from Steven Hughes' cross and powered into the bottom corner through a crowd of players.
But Alexander's first, a simple tap-in, changed Jackett's teamtalk for the better.
Saddlers stopper Gilmartin could only parry influential Mark Laird's side-footed drive into the path of Alexander, who simply couldn't miss.
His second, which gave the hosts the breathing space they needed, showed the striker at his best.
He bombed into the Walsall box to meet Martin's in-swinging cross which Anthony Gerrard half-cleared with his header.
Whitbread was on hand to keep the move alive and nodded the ball back across for Alexander to hammer home on the turn.
The game was virtually over as a spectacle with the Lions fans cheering passes as the hosts enjoyed almost total domination.
There was a moment of real concern midway through the half when Walsall defender Emmanuele Smith was stretchered off unconscious.
Lions keeper David Forde landed on the chest of the young defender as the pair clashed at a 68th minute corner.
The game was stopped for 10 minutes as he received treatment before being carted off to hospital for x-rays.
The players seemed unable to recover from the pause and the game failed to get going.
Lewis Grabban had a muted penalty appeal waved away by referee Clive Penton as he fell in the box in the first of the eight minutes of injury time, while Deeney and Ibehre remained the most lively of the below par Walsall outfit.