First-half goals from Teixeira and Scottish international striker Craig Beattie earned Albion a third successive home win and extended their 100 per cent record in front of their own fans this season.
But it was frustrating trip to The Hawthorns for Simon Davey's Barnsley side, who were hammered 7-0 by Albion on their previous visit on the final day of last season.
Once midfielder Teixeira, who cost Albion £600,000 from Academica de Coinbra in July, opened the scoring the result was never really in doubt.
But Barnsley can still take heart from a spirited display that was epitomised by the non-stop display of skipper Brian Howard.
They enjoyed a purple patch towards the end of the game when Anderson De Silva forced Dean Kiely into a smart 75th minute save before Grant McCann fired wide 60 seconds later.
But by then the damage had been well and truly done by Albion. Teixeira, who had threatened as early as the fifth minute, gave Albion a deserved lead on the half hour.
He capitalised on a neat pass from Kevin Phillips and cut in-field before firing a low drive beyond Heinz Mulller from the edge of the area.
Barnsley were still coming to terms with that setback when Beattie doubled Albion's advantage.
A long through ball from Paul Robinson released Beattie. The Scottish international beat the offside trap and coolly drilled his shot past Muller.
Barnsley struggled to really test Albion goalkeeper Dean Kiely in the first half with Kayode Odejayi wasting one golden chance after 11 minutes before Anderson De Silva fired just wide in the 35th minute.
In truth, Davey's side were fortunate to get to half-time only two behind as Phillips crashed a header off the bar in stoppage time before Muller tipped another effort from Beattie over the bar.
Muller also denied Zoltan Gera in the 55th minute and substitute Ishmael Miller after 77 minutes before skipper Jonathan Greening smashed a drive against the Barnsley bar in stoppage time.