Java (OpenJDK 8), 70 bytes
a->{int i=1;int[]c=new int[a.length+i];for(int u:a)c[i++]=u;return c;}
Explanation:
int i=1; //Initialize counter
int[]c=new int[a.length+i]; //declare new array with length: 1 + input array length
for(int u:a) //Loop through each element in input array
c[i++]=u; //Increment counter after shifting each element up one spot in the output array
return c; //return new array starting at index 1