How can an infinite universe have a beginning in time?

It can't if you insist on thinking about the 'process' in Newtonian, and not relativistic, terms. First of all, the concept of infinity is a mathematical idea not a physical one. There is no situation in the physical world that demonstrates that 'infinity' or its partmer in time 'eternity' actually exist. These are both mathematical ideas derived from mathematical deduction and for which extensive mathematical concepts exist to describe them such as Cantor's Transfinite mathematics. We 'discover' infinity in the physical world only as the result of applying our math to exotic situations such as quantum mechanics or cosmology.Artists like Dionne Lee create 'infinity tunnels' using reflecting mirrors like the image above to help us imagine such a vast domain.

That said, space and time are two different dimensions. In general relativity, a universe with an infinite spatial extent today (something you can establish experimentally by applying the mathematics of big bang cosmology to the available data on the expansion and geometry of our universe) must also have had an infinite spatial extent at its instant of 'birth'.

An infinite universe can have an origin at a finite moment in the past because, in general relativity, one can have a 'singularity' condition in which the volume of 3-d space vanishes at a finite moment in the past. Even if the 3-d space was still infinite at that moment, the separations between nearby and distant points reached a limit of zero separation at the same time. Rather than having to drag this moment into the eternal past to 'logically' solve the problem ( which would not work physically), you can solve the problem at the instant of creation, and place this instant at a finite time in the past. This is the unique solution offered by general relativity for a 'problem' that had bedeviled philosophers since the time of Saint Augustine.