Pitman smashed home a 40th-minute free-kick from 25 yards with Crawley's defensive wall in apparent disarray.
Bournemouth have now gone 15 matches unbeaten since Eddie Howe returned to the south coast as manager from Burnley and moved up to fifth place in the League One table.
Assitant boss Tindall said: "That was a great piece of quick-thinking by Brett. The referee asked him if he wanted to take it and he did.
"Brett's goal came at the right time and gave us a two-goal lead at half-time and set us up for the second half."
Crawley were struggling from the 16th minute when their striker Gary Alexander headed Steve Cook's cross past his own goalkeeper.
Pitman's goal came after Marc Pugh had been fouled on the edge of the area and there was no way back for Crawley once Probert let it stand.
Barker's team were much-improved after the break and only a crucial save with his legs by Bournemouth goalkeeper David James denied substitute Billy Clarke a 70th-minute goal, but Howe's side made sure of the points in the 84th minute when Simon Francis and Lewis Grabban set up a chance for midfielder Eunan O'Kane to run clear and score.
Tindall added: "The first goal was vital and the second gave us a bit of a cushion. The second-half performance was very professional because we knew Crawley were not going to lie down.
"We have played some great football in the 15 games we have been here. Our movement, rotation and overall football has been excellent."