This article focuses on how collective improvisation, as a play with situational affordances and constraints, can facilitate or elicit luck in social innovation processes. We propose the improvisation perspective to offer a means to move from ungovernable to governable serendipity in innovation. Furthermore, the article presents Trasformatorio, an innovation methodology based on the living lab approach, that takes innovation-through-improvisation as its premise. Following an analysis of Trasformatorio's social innovation efforts, we conclude that while governing serendipity may be a challenge, improvisation provides an opportunity to innovate responsibly. Improvisation refocuses ideas about unexpectedness and anchors the unforeseen to a process approach. This, in turn, leads to insights about how responsible social innovation can be governed by means of (1) situation awareness; (2) collective brokering; and (3) explicit reflection about how product and process, here conceptualised as part of a Design Space and Narrative Space respectively, interrelate.