Pulis played under Redknapp at Bournemouth during the 1980s and would have been delighted to pick up a point in his first game in charge.
The scoreline also gave Saints their fifth straight draw in the Championship this season.
The home side started with more purpose when on three minutes Tomasz Hajto crossed to fellow Pole Kamil Kosowski who headed low towards the Plymouth goal. However, Romain Larrieu produced an excellent save to tip the ball around his left-hand post.
It was Kosowski who was involved with the vast majority of Saints' attacking play and he was unlucky when after racing past Anthony Barness he was cynically brought down by Plymouth captain Paul Wotton just outside the penalty area. The ensuing free-kick however failed to trouble Larrieu.
Plymouth, roared on by a fanatical away following of 2,500, could only muster one opportunity in the first period when former Saint Micky Evans intercepted a Danny Higginbotham pass and drilled wide of Antti Niemi's goal.
The best chance of the half fell to Brett Ormerod when Ricardo Fuller caused some disruption inside the Plymouth area and when the ball fell to the striker he seemed to take an age before shooting straight at Hasney Aljofree.
Some rash challenges in the second half seemed to spice up the occasion for both the crowd and the players and Fuller was guilty of missing the best chance of the game when he fired wide after Nigel Quashie had released him inside the area.
Saints boss Redknapp threw caution to the wind with 20 minutes to go by introducing youngsters Kenwyne Jones and Theo Walcott but neither of them could inspire the home side to snatch the victory.
And it was no surprise when the loudest noise made by the home following was the chorus of boos which greeted the final whistle.