The option terseinits does that for you. terseinits is a meta-option that essentially executes the following definitions
\renewrobustcmd*{\bibinitperiod}{}
\renewrobustcmd*{\bibinitdelim}{}
\renewrobustcmd*{\bibinithyphendelim}{}
and sets the test \ifterseinits (that test is not used by a lot of styles, apparently, so it hardly matters).
If you want a more fine-grained control, you can redefine these macros yourself. They do pretty much what their names suggest: \bibinitperiod is the punctuation after a name initial, \bibinitdelim the space between two name initials and \bibinithyphendelim replaces the two between hyphenated name parts such as Jean-Jacques.
\documentclass{article}
\usepackage[backend=biber, giveninits=true, terseinits=true]{biblatex}
\DeclareNameAlias{author}{family-given}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{key,
author = {Lastname, First Second},
year = {2001},
title = {Title},
publisher = {Publisher},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\cite{key}
\printbibliography
\end{document}

Note that I changed the deprecated last-first to the new family-given, since you are already using the new name giveninits (cf. Biblatex 3.3 name formatting).
If you are using authoryear, you may want to redefine sortname and not only author:
\DeclareNameAlias{author}{family-given}
biblatexsubmissions.biblateximposes quite a different workflow on publishers than standard BibTeX orthebibliography, plus publishers are known to prefer stable (read: older) systems, where incompatibilities with modernbiblatexcould arise. Many journals have LaTeX templates and those almost never featurebiblatexand usually insist on particular.bststyles orthebibliography. See also https://tex.stackexchange.com/q/12175/35864 – moewe Mar 28 '19 at 08:52