Taylor scored the decisive free-kick to seal a 5-4 penalty shootout win at Bramall Lane, after a 2-2 draw in which the visitors twice led. Taylor had earlier put the Brewers ahead in extra-time.
"We always knew it was going to be a great start to the season and we've told the players that they've done the work, so they should relish it," said Burton boss Gary Rowett.
"I said to them before the game, whatever happens, just react in a positive way and deal with the problems."
Crewe Alexandra registered one of the most impressive results of the day, demolishing League One rivals Hartlepool 5-0 at Gresty Road.
"I am delighted with the start we have made, but we are not getting carried away as there is still a lot of work to do on the training ground," said Steve Davis, the Alex boss.
“It's just the start of a long hard season, but it's one we can all look forward to after that. Our front four are going to be difficult to hold for any defence at any level, they have movement and power and score goals."
Bury caretaker boss Peter Shirtliff's first game ended in a 2-1 home loss to Championship side MIddlesbrough.
MK Dons needed penalties to get past Cheltenham, winning the shootout 4-2 after their tie finished 1-1 after extra-time at Whaddon Road.
Carlisle and Walsall both progressed, thanks to 1-0 home wins over Accrington and Brentford respectively.
The former Iron striker Chris Dagnall hit a brace as Barnsley edged past League Two side Rochdale at Spotland, registering a 4-3 win after extra-time.
Wolverhampton Wanderers, Leeds and Hull City also progressed on Saturday.
The first round of the competition was due to be played over the coming three days, with the Iron heading to Derby County on Tuesday evening.