The following code builds a string containing the current transaction and either prints it or throws it away depending on the state of a keep flag.

use strict;
use warnings;

my $numbers = join '|', qw(2345678923 2121212121 4424352424 2323232323
+);

my $transaction = '';
my $keep;
my $startRe = qr!<BEGIN Transaction>!;
my $endRe = qr!</END Transaction>!;
my $match = qr!(^|\D)($numbers)($|\D)!;

while (defined (my $line = <DATA>)) {
    my $inTransaction = $line =~ $startRe .. $line =~ $endRe;
    next unless $inTransaction || $keep;

    $transaction .= $line;
    $keep ||= $line =~ /$match/sm;
    next unless $inTransaction =~ 'E0$';

    print $transaction if $keep;
    $transaction = '';
    $keep = undef;
}

__DATA__
<BEGIN Transaction>
2345678923
</END Transaction>
<BEGIN Transaction>
1
</END Transaction>
junk
<BEGIN Transaction>
12345678923
</END Transaction>
<BEGIN Transaction>
2121212121
</END Transaction>
[download]

Prints:

<BEGIN Transaction>
2345678923
</END Transaction>
<BEGIN Transaction>
2121212121
</END Transaction>
[download]

Note the use of the flip-flop operator (..) to keep track of inside/outside the transaction and to detect the last line of the transaction.

Note also the use of qr to precompile the regular expressions that are used for matching transaction start/end and (more importantly) to match the numbers.