Did you use warnings;? When I do, I get warning messages:

Unrecognized escape \w passed through at ./683940.pl line 6.
Unrecognized escape \d passed through at ./683940.pl line 6.
Unrecognized escape \d passed through at ./683940.pl line 6.
[download]

This is my version of your code:

#!/usr/bin/env perl
use warnings;
use strict;

my $NetName = 'top_level/mbist_wrapper_pix2d[22]';
my $NetPattern = "mbist[\w+\d+\/]*\[\d+\]";
if ($NetName =~ qr/$NetPattern/x) {
    print "match\n";
}
else{
    print "no match\n";
}
[download]

One solution is to avoid interpolation (as jettero has already mentioned).

#!/usr/bin/env perl
use warnings;
use strict;

my $NetName    = 'top_level/mbist_wrapper_pix2d[22]';
my $NetPattern = 'mbist[\w+\d+\/]*\[\d+\]';
if ($NetName =~ qr/$NetPattern/x) {
    print "match\n";
}
else{
    print "no match\n";
}
[download]